oh let's do some algebra review solve 4K plus 7 parentheses negative 4 minus 2K NBA video game uh equals negative two parenthesis K minus two well whenever we have parentheses on each side of the equation we should probably distribute right off the bat so let's do that 4K is not affected uh 7 times negative 4 is negative 28 7 times negative 2 K is negative 14 k that's going to equal distribute again negative 2 times K is negative 2K that's a lousy basketball game joke it's great basketball game negative 2 times negative 2 is positive four all right after we distribute we should probably try to combine like terms on either side so I have 4K over here I have minus 14k and that gives me negative 10K nothing happens with negative 28. nothing happens with negative 2K and nothing happens with 4. I have K on both sides of the equal sign I have a negative 10K over here and a negative 2K over there it doesn't really matter which one I get rid of but I always like my K ending up on the left so let's do the opposite of this guy and add 2K to both sides like so negative 10 K plus 2K is negative 8 K drop down the minus 28 drop down the equals drop down the positive 4. now it's a two-step equation okay in order to get K all by itself negative 8 is attached but the first thing we want to do is do the opposite of negative 28 and add 28 to both sides like that that gets us negative 8 k equals 32. one more step in order to get K all by itself we do the opposite well not the opposite we do the inverse of negative eight which is divide negative eight divide negative eight and K is going to equal negative 4. so why are we starting a geometry series by doing algebra you're going to see a lot of algebra specifically numbers and variables on both sides of an equal sign so if you thought algebra was going to go away for geometry no like And subscribe all right I have to simplify 3 times the square root of 405. ladies and gentlemen I'm going to use the Nick parich method of simplifying square roots step number one take your number inside the square root and make a factor tree out of it so 405 I can divide that by 5 and that gets me 81 I think Circle any prime number what two numbers multiply out to 81 9 and 9. can you Circle either those nope because they're not prime but I'm going to be able to Circle A lot of these 9 is 3 and 3 circles Circle nine is also three and three circle circle so the prime factorization of 405 is 5 times 3 times 3 times 3 times 3 but not in that order let three still hang out on the outside there let's write these in order three times three times three times three times five what you do now is since this is a square root we're going to circle sets of identical pairs like this group of Threes And this group of Threes don't worry about that five it doesn't have a friend kind of like me in high school he he full of jokes bring out one three bring out another three and whenever we bring out more than one thing we're going to end up multiplying so 3 times 3 um times the 3 that's already out there square root you're now gone you're now gone 5 is left over so when I put all of this together 3 times 3 times 3 is 9 times 3 which is 27. what's left over root five and that's my guy so step one make a factor tree step two listed out an order step three Circle groups of pairs step four pull out one of them step five multiply the Nick Parish method of simplifying square roots patent pending oh let's simplify that root minus that root minus that root well you treat square roots as if there were variables and if they were the same variable then you would combine like terms and that would be easy well none of these are the same so you might be thinking oh great we can't combine like terms we're done wrong you can combine these we just have to simplify all three of these square roots so in a previous video I did the Nick parich method of simplifying square roots where I took all three of these and made a factor tree for them I'm not going to do that much I'm just going to use a little bit of common sense so let's draw this out in Black 12 can be written as 3 times 4 which is two times two times three and then we have minus 2 7 can be written out as 3 times 9 or 3 times 3 times 3 prime factorization again if you need to make a factor tree go right ahead 45 is 9 times 5. which is three times three times five now that we have the prime factorization written out Circle Pairs and bring one of them out Circle Pairs and bring one of them out so Circle Pairs and bring one of them out so I have negative three times two and what's left over is regular three minus two I brought out a 3 so times three and what's left over is the square root of three minus two bring out a three what's left over is the square root of five so negative 3 times 2 is negative six negative six root three negative two times three is negative six negative six root three negative two root 3 is negative six negative six ooh root five so now what I have to do is combine things that have the same exact square root negative six root 3 is literally the exact same thing as negative six root three so if I take away six root 3's and I take away six more root threes I'm taking away twelve root threes Okay negative six minus six is negative twelve root 3 comes along for a ride well what about this guy this guy wants a ride too can I join in no because you're different and that's how we roll in math we segregate and we don't care about others that's really a terrible thing but since these guys aren't the same we can't combine them so that's my answer I'm done simplify the square root of 15 times parentheses root 3 plus root 10 close parentheses well Common Sense tells us that we should do good old distributive property so root 15 times root 15. remember when you multiply two square roots you're allowed to multiply them within one square root so what that means over here is square root of 15 times the square root of 3 is the same thing as a square root of 15 times 3. which is 45. distribute again the square root of 15 times 10 is the square root of 150. now before you Circle that and call that a day we're probably not done well we're definitely not done these actually might be able to combine maybe maybe not I don't think so but it's possible and what we need to do is we need to take these aside and simplify these and see if they end up being the same square root if they're not then you know we just have a more simplified version so if I wanted to take 45 and make a factor tree out of that that would be 9 five three and three and so using the Nick parich method of simplifying square roots this ends up being three times three times five doing the same thing with 150 150 is 10 and 15. nothing happens there ten is two and five Circle them both fifteen is three and five Circle them both and plus 150 within a square root is going to be two three five five two times three times five times five now the next step is for each one individually these Circle groups and bring it out so I Circle a 3 bring out one of them and what I have left over is the square root of five I Circle fives bring out one of them so plus five and what I have left over is two times three which is six so although I wasn't able to combine these and I kind of knew that would happen I still did my multiplication and I still simplify it as best as I could okay so that's multiplying square roots baby simplify the square root of 15 over the square root of 6. two things happen here thing number one is if I give you something like the square root of a over the square root of B I'm allowed to write that as the square root one big square root of A over B I'm allowed to do that why am I going to do that because I can write this out as the square root of 15 over 6. and I know that 15 over 6 simplifies to 5 over 2. I could divide the top and the Bottom by three great does that mean I'm done no let's write it back to the way it kind of once was where we have a square root over a square root there's some stupid math rule that I still don't understand why even though I've been teaching math forever that says you're not allowed to have a square root on the bottom you can't do it so what we are going to do is a process called rationalizing the denominator in order to make that square root go away what we can do is multiply the Bottom by the same exact thing if I have a square root of 2 and I multiply it to the square root of 2 the square root of 2 times itself is the square root of 4 which is 2. however we know that when we deal with fractions we can't just multiply it by a number we have to multiply it by a number that equals one so I'm going to multiply the top and the Bottom by the square root of 2. that makes this root 5 times root 2 which is the square root of 10 over again I'm going to explain it and I'm going to show you what I did the square root of 2 squared okay root 2 times root 2 is root 2 squared now a root 2 squared means that the root and the squares cancel each other out leaving us with the square root of 10 over 2. now before you start thinking oh great this could become five nope the square root of 10 is a square root the Whole Thing Lives within this square root 2 is no longer a square root so these guys have nothing in common with each other now and you can't just simply divide what I do have is I have a fraction where I don't have a square root in the denominator anymore which means it's a legal problem okay and there you have it dividing square roots fun the segment addition postulate the sediment Edition postulate says if I have a bunch of segments stuck together that you can add those segments up and it'll equal the whole thing hmm so let's see what I know I'm supposed to find i k I'm supposed to find from here let's go to open from here to here I know you're 12 but I don't know KL oh but I do know that J K plus k l is 31. which means if I want to find KL here I could do 31 minus 12. 31 minus 12 is 19. okay so let's get rid of that question mark now you might be thinking uh that's not going to help me because I need i k not k l I know but I'm working towards something here I'm working towards something here what I also know is i l is the whole thing is 49. I know that J K and K L add up to make 31. so if I wanted to find out what i j was I could do 49 minus 31. which is 18. in a way it was kind of useless it was kind of pointless to find KL because eventually I didn't really need that but you know I had fun doing it so it's too late now am I done no because I have to find i k still haven't found that yet ah but i j is 18 and J K just kidding is 12 and I K is 18 plus 12. so i k is thirty oh 30. that's going to be my age in a few years just kidding and that ship has sailed a long long time ago but I still sound great like And subscribe foreign points a B and C are collinear that means they're all points that live on the same line so let's draw a fake line just for funsies B is in between a and C another fake line with points now solve for x all right what do I know I know AC is 22. so I know this whole thing oh my gosh it's 22. that looks great 22 the Taylor Swift special I know that BC is X plus 14 b c of course standing for it before Christ uh and a b is X plus ten so what do I know well according to the segment addition postulate a B plus B C equals AC which means a b X Plus 10. plus BC so I can put that in parentheses even though it's useless plus X plus 14 is going to equal 22 the aforementioned Taylor Swift special now these parentheses were useless so let me rewrite it getting rid of those parentheses I put those parentheses there just because when you substitute you technically should have parentheses sounding like a teacher let's combine the x's on the left x and x make 2X 10 and 14. make 24 and that equals 22. now it's a two-step subtract 24 from both sides so that leaves us with 2X equals negative 2 one last step divide by 2 divide by 2 and x equals negative 2 over 2 is negative 1. now you might be thinking no no no no no you can't have a negative length this isn't a negative length this just gives us an X so if I wanted to find what these lengths are I could be like well this is going to be negative 1 plus ten nine and this is going to be negative 1 plus 14 13 and 9 plus 13 is 22. so weird answer especially for geometry when you want to stay away from negatives but this is the answer and I'm sticking to it find d e which is this chunk right here this uses the segment addition postulate which says if I add you and add you and add you I get this whole thing so since I'm given information about each little segment here let's just do that let's add them all up d e is three x minus 28. plus e f is three x minus 30. Plus FG which is just X and all of that equals 33. if you have nothing in front of a parentheses you don't need the parentheses so this leaves me with 3x minus 28 plus again nothing in front of a parentheses 3x minus 30 plus X equals 33. let's combine some like terms we got 3x we've got another 3x and we've got 1X so that's 7X we've got negative 28 and negative 30 which is negative 58 and that equals 33. now it's just a regular two-step equation let's add five eight fifty eight I don't know why I said 5 8 but I did I'm I'm not changing anything 7x is going to equal 91. last step since 7 is attached to X let's detach it by dividing everything by 7 and 91 divided by 7 is 13. now before I Circle that and be like dunzo I'm not it didn't ask me to find X it asked me to find d e but the only way that I could find d e is by finding X and now plugging it into that guy right there so d e is 3 times x 3 times 13 minus 28. running out of space so 3 times 13 is 39 30 min my gosh 39 minus 28 is 11. so d e is 11 no units to worry about no centimeters no inches just 11 baby we found it yeah classify each angle as acute like me obtuse right like me or straight no comment I have an acute angle and acute angles mean they have to be less than 90 degrees not less than or equal to less than 90. obtuse has to be an angle that's going to be in between 90 and 180. right is always 90 on the DOT straight is 180 on the DOT so 180 straight when I draw an A line that's 180 degrees it's going to look like that okay now 97 is going to be like a right angle but just a Teensy bit more that's obtuse because it's more than 90. 39 if I were to draw an angle uh half of a right angle is 45 so if I just kind of keep it close that's what 39 would kind of look like I can get a protractor and do all this but I'm not going to all I have to do is just say that it's acute because it's less than 90. you're greater than 90 smaller than 180 so you're obtuse you're clearly less than 90 so you're acute and those are my Five Guys like the burger place name the angle in four ways okay when you name an angle you have to figure out what your vertex is first your vertex is where the two rays that create your angle meet one of your angle names can be the angle f okay just f as long as you use your vertex you're not allowed to say angle G you're not allowed to say angle e but you are allowed to say angle F since it gives us that number here I'm allowed to call it angle four now there's two other ways that you can name it we include points G and E along with this so one way that you could also name this is write this out this g f e if you use three letters you must always have your vertex in the middle of those three letters and if you're thinking well why did you start out with G couldn't we start out with e yup as long as you put F in the middle it doesn't really matter so all four of these are ways that you can name this angle right here what's key is noticing that your vertex is f everything is pretty much based off around it we also had a number that works too this means the measure of angle ihq so find the measure of angle ihq if the measure of IHG is 176 degrees and the measure of qhg is 130 degrees this is going to use What's called the angle addition postulate the angle addition postulate says if I have two angles like this one right here and this one right here glued together I can add them up and equal that whole thing right there what I know is i h q i h q is uh I don't know actually that's what I needed to find so we'll call you X i h g let's use a different color IHG is this whole thing so i h g is a hundred and seventy six degrees q h g q h g this guy right here is 130. so X Plus 130 is 176. X X Plus 130 is 176 why because the two smaller angles add up to make this big green angle here all I have to do for this guy is subtract 130 degrees from both sides and X is going to equal 46. my current age now I'm not going to call it X I just used X as a variable so what I'm going to do is go back to the original thing that had asked me to find the measure of angle i h q is 46 degrees okay angle addition postula with really basic numbers nothing crazy no wacky problems you could solve this without using an X you could solve this using common sense just by saying oh isn't it just 176 minus 130 yep but I got a little bit more involved because more time doing these problems means more YouTube money for me and I'm all about lining those Pockets baby that's why I keep talking at the end of these videos sometimes maybe those precious extra seconds will help feed my family also I'm not 46 I'm a little younger than that a little like And subscribe the measure of angle g f n e or X Plus 10. 10 the measure of angle n f e not to be confused with nfts is 14 X plus three the measure of angle GFE is 157 find the measure of angle n f e all right let's label some stuff gfn g f n g f n is 4X plus 10. so let's sneak that in there okay uh n f e thank goodness is going to be 14 X Plus 3. g f e g f e this whole the whole thing is 157 degrees which means I can set up an equation that says 4X plus 10 4 X plus 10 plus 14 X plus three equals 157 why did I put those parentheses there just because you should always put parentheses when you substitute even though in Geometry more often than not you don't need them but one of these days it might be needed so the parentheses are there although since there's nothing in front of the parentheses or just a plus sign in front of the parentheses we don't need them so I'm getting rid of them okay so now we have ourselves a pretty gross looking equation but I do have stuff on the left like 4X like 14x that combine to make 18x I have 10 and 3 which I think is 13. and that equals 157 still now it's a two stepper in order to get X all by itself we have to get rid of 18 but let's get rid of 13 first by subtracting 13 from both sides like so 18 x equals 144. divide both sides by 18. and that's going to end up being 8. of course I did that off my head and I didn't pause the video so I could ask Siri to do it for me X is eight are we done no it's not asking me to find X it's asking me to find n f e n f e is 14x plus 3. so the measure of angle nfe is going to be 14 times x which is now 8. 3 14 times 8 is of core 112. again I didn't ask Siri and pause the video I just did it in my head 112 plus 3 means the measure of angle n f e I keep wanting to write NFL but I didn't it's 115 by the way the NFL does not sponsor any of my math videos but if they're interested just let me know NFL let me know Goodell I'm a big Eagles fan so just give me a call find X what do we have here we have a picture where I have two angles that are across from each other oh these are called vertical angles when two angles are created by you either crossing over like so angles that are opposite of each other are called vertical angles and vertical angles equal each other so in this case 2 plus 3 x is going to equal 62 because these guys are vertical angles okay uh uh so let's get rid of the parentheses let's add some color and get rid of the parentheses don't need the parentheses I just put them there because they were over here I have to get X all by itself 3 is attached to X but let's get rid of two because it's floating cross you out 3x is equal to 60 now now it's a one step in order to get rid of three we have to divide three from both sides like so when X is going to equal 20. and that's all we had to do just find X it's not really necessary to put a parenthesis or a degree symbol there just because X represents a variable and not a measurement but you know if you did I don't think your teacher would Mark you wrong I certainly would you might have a mean teacher that would but X is 20. oh speaking of uh 20 that's that's how many years old I was a few years ago ah [Music] foreign X where this angle right here is 29 this angle right here is x minus 24 and this huge angle is 2 96. now do these guys add up to make 90 no do these guys add up to make a straight angle no so not 180 uh do these guys equal each other no I don't think so it doesn't look like they do what do we know well they had to give us that 296 for some special reason and that's because the only way we're going to do this is the fact that an entire circle an entire angle spun around beginning to end is going to be 360 degrees so my job is to find X where I know that this angle is x minus 24. I know that this angle is 29 so we'll add 29 the rest of the circle is going to be 296 and that whole thing is going to equal 360 degrees so that'll help us find our X so let's get rid of the parentheses don't need them x minus 24 plus 29 plus 296 equals 360. negative 24 and 29 make 5. so plus 296 comes along for the ride 360 comes along for the ride these guys combine to make 301 so 360 is still living there I now have a one step equation where all I have to do is subtract 301 subtract 304 and x equals 59. 9 59 . so there you have it that is the angle addition postulate in action fun assume the lines are parallel and let X that guy equal 59 degrees classified didn't mean to do that X and Y then find all of the angle measures well here's X here's why they are related because they are considered chorus bonding angle corresponding angles are angles where if I were to take this rip it off and put it down here it's literally the exact same Angle now by saying they're the exact same angle since X is 59 degrees that means Y is also 59 degrees now X and this guy right here are vertical which means that 2 is 59 degrees and Y in this guy right here are vertical so that means they're 59 degrees so that makes life a little easy to remember the other thing too is if this is 59 degrees and I want to find these angles Y and this angle right here are glued together to form a straight angle which means they both add up to 180 degrees so in order to find all the other angles 180 minus 59 is going to give me 121 degrees so you're 121 degrees which means its vertical angle is 121 degrees this guy and this guy are considered alternate interior angles which means that's 121 degrees and these guys are vertical making them 121 degrees so whenever you have parallel lines like so and you're being asked to find all of the angles whatever this guy is three others are going to be that and whatever this guy is the other three are going to be that so there you go solve for x then find each angle measure all right so we have parallel lines that's pretty important we have an angle on the very top left and an angle on the very bottom right now these two angles have a name as far as the way they're related and they're called alternate exterior angles when given parallel lines alternate exterior angles are congruent which means I can say negative 1 plus 14x that's the guy on the top left is going to equal 12x Plus 17. and now it's just a equation with X on both sides of the equal sign problem so I'm going to subtract 12x from both sides like so minus 12x minus 12x and across you out that leaves me with negative 1 plus 2X equals 17. now it's a two-step equation so let's add one let's add one cross U out 2x equals 18. divide by 2 divide by 2 x equals 9. now I did part one which was solve for x part two has me finding each angle measure well just because X is nine doesn't mean the angle measure is nine in fact it's not the angle measure could be U or U as long as I plug in 9 for it so why don't I take this one 12 times 9 plus 17. 12 times 9 is a hundred and eight plus seventeen is going to be 125. so this angle down here on the bottom right is 125 degrees which means it's alternate exterior angle is also going to be 125 degrees if I were to look at this angle and go directly opposite of it I have vertical angles which means that's 125 degrees and this angle is alternate interior to that angle which means you are 125 degrees it just makes sense because it's vertical to that guy right there now if all of those are 125 degrees I have to keep in mind that if I were to take any two adjacent angles in any of these problems here any angle glued to each other is going to add up to make a straight angle in other words 180. so if I wanted to find out what all of these angles are I'm going to do 180 minus 125 which gives me 55 degrees so your 55 degrees which means its vertical angle is 55 degrees which means its alternate interior angle is 55 degrees which means its vertical is 55 degrees so when you're given parallel lines and you're being asked to find angle measures it's either going to be one angle and it's supplement adds up to 180. and if it's if whatever angle you come up with three other angles are going to be the exact same thing vice versa with these guys so there you go fun fun plot each point on this coordinate plane right here well I have a good chunk of the alphabet so I'm just going to go and get started remember when you graph a point it's given to you in X Y form X when it's positive goes right when it's negative goes left why when it's positive goes up and when it's negative goes down so to graph a which is 710 I go right one two three four five six seven up one two three four five six seven eight nine ten and year a B is don't move go up four so start at the origin don't move up one two three four B C is go oh it's a negative which means I go left one up ten so all the way up to 10. that's my C D is go left one two three four five six go down one two three four five six that's d e is go right ten and then don't move so write 10 don't move that's e f is go right nine up seven so right nine up seven I think I counted that off correctly that's my f G is left three down four so one two three down one two three four that's my G H is Left four down nine so one two three four down one two three four five six seven eight nine that is my h I is 4 1 so right one two three four up one that's my I and J is right one two three four five six seven down nine so seven negative nine lives right there that's my J so quadrant one pretty well represented quadrant two not so much quadrant three quite a few dots quadrant four not so much I had a DOT on the x-axis and a DOT on the y-axis so that's how you graph points on a coordinate also known as a Cartesian plane of course points half of the name of this video series that I've created just for you like And subscribe State each coordinate uh let's go in alphabetical order now we know before I go crazy here we know that a point is given to you as an X and A Y positive X means you have gone right Negative X means you have gone left positive y means you have gone up negative y means you have gone down so whenever I look at my points I'm going to be like well how did I get from 0 0 to there left up would mean negative positive so let's get that out well I'll leave that there for a few minutes I'll leave it there until I can get rid of it so a was left all the way to 10 up to so negative 10 positive 2. B B is here you are right one two three four five six yup up all the way to ten so right 6 up 10 is positive 6 positive 10. C where are you here's c c is left eight down three so C is left eight which is negative eight down three which is negative three now I'm going to get rid of U and don't forget the parenthesis parentheses parenthesi parenthesi PSI Gangnam Style remember him made K-pop famous D there you are left eight down eight so left eight is negative eight down eight is also negative eight e is here so right one down five so e is right one which is positive one down five which is negative five f is right next to it so you go right two down one two three four so f is right two positive two down four negative four G is right here so G is left seven up eight so left seven is negative seven up eight is positive eight H is right two up five so H is positive two positive five I is here so right eight down two so I is eight negative 2 J there you are J is left seven down five so J is negative seven negative five okay am I done am I done one two three four five yeah I'm done oh I did it good job everybody good J for job hmm State the quadrant or axis that each point lies on well let's draw ourself a little coordinate plane and state that the Quadrant One axis lives on the top right and then you go counterclockwise so quadrant two quadrant three quadrant four now when you're looking at points quadrant one has me graphing by going right one up one or right however many up however many and so that's going to be positive positive positive X positive y if I go left up that's quadrant two going left is a negative x value but going up is still positive y going left down puts me in quadrant three that's a double negative in quadrant four has me going right which is a positive X down which is a negative y so T is negative 3 positive 5 has me going left three up five quadrant two U has me going right one don't move so going right one don't move doesn't put me in any quadrant but instead instead having me live on the x-axis so one zero is on the x-axis something like 0 1 would be the y-axis but I didn't ask you that see what I did I used slang negative 5 positive five is the same as this negative positive negative positive quadrant two again what so those are your quadrants those are your axes and those are the points that live with them the end find the midpoint now we're doing this graphically the midpoint is exactly what it sounds like it's a point in the middle of a line segment we have an end point here at negative 4 1 and we have an end point here at 1 negative four all right so the distance between this guy to this guy is four so the midpoint is going to be somewhere in the middle of it around here this goes from positive one to negative 4 which means the distance here is five so somewhere in between is going to be my midpoint now we like to call midpoint m and it looks like if I were to list out what this point is I would go negative 1 to get there down one and a half so my midpoint is going to be negative one down negative one and a half now if you're thinking oh you sounded a little insecure about that sir is there a way we could do that algebraically like math and stuff there sure is that's the next video make sure you watch that too but this is just graphically and we were able to look at the middle of everything by kind of eyeing it up and that looks about right to me because it is right so there's your midpoint find the midpoint no graph and instead I have 2.52 negative 4 negative 3. the midpoint formula says let's take one of these points doesn't matter which one's which and call it X1 y1 let's call the other point x sub 2 y sub 2. the midpoint formula is this it's a big point and the x value is going to be x sub 1 plus x sub 2 all over 2. and if you're thinking that's like the average it is like the average average means middle okay um y sub 1 plus y sub 2 all over 2 gives me uh the midpoint for the Y value now it's just a matter of plugging stuff in and doing a little bit of math all right so x sub 1 is 5. plus x sub 2 is negative four all over 2 y sub 1 is 2 plus y sub 2 is negative 3. all of that over too 5 plus negative 4 or 5 minus four is one so one over two two plus negative three or two minus three is negative one over two so we'll slide that negative out there because I don't have enough space to draw an under another another thing underneath and so that's my midpoint you know the kids tell me that this is really mid and I'm like it sure is uh mid means like average I guess I don't know I'll check Urban Dictionary like And subscribe find the distance between these two points all right so I got that point right there I've got that point right there I really can't like get out of ruler and measure out how long that is it's neither horizontal it's neither vertical plus I don't want a ruler because Jesus is the only ruler that I care about or the president or the king of England but more Jesus so what I can do is I can do this I'm going to draw a perfect line as you can see me making right there and right there and what I've just created is a poorly drawn right triangle I know that this length is one two three four five six seven eight nine spaces long and I know that this one is one space long and if I'm trying to find the missing side of a right triangle I'm going to use What's called the Pythagorean theorem right there there you go now what's a what's B and what's C well in a right triangle the longest side is always going to be C and the longest side is always going to be the one opposite of the right angle which ones a which one's B doesn't matter as long as the longest side is C because you're just going to square them and add them anyway so let's call 9A let's call 1B so when I plug it in I have 9 squared plus 1 squared equals c squared I have to get C all by itself but first I can do a little bit of PEMDAS over here so 9 squared is 81 1 squared is let me get out my calculator 1 times 1 is 1 and then that still equals c squared uh 81 plus 1 is 82 that equals c squared square root both sides remember if I want to undo a square I square root both sides and so C equals the square root of 82. now depending on what teacher you have they might be like well I need you to find out the decimal to the nearest whatever so the square root of 82 is like nine point something uh if you have a teacher that says you need to break this down and turn that into something times the square root of something you can't do anything here 82 is 41 times 2. you can't break that down break that down any further and bring stuff out so we're done C is the square root of 82 or 9 point whatever it is but that's what we did we used the idea of a right triangle to find out the distance of that dot to that dot which ended up being a little over nine which makes sense dollars and cents find the distance between the two points eight five and negative one three now I could graph that and do what I did in a previous video and make a right triangle but I ain't I'm going to use What's called the distance formula the distance formula says I'm going to take a big square root I'm going to do in one square root x sub 2 minus x sub 1 square it I'm going to add y sub 2 minus y sub 1 square that that's the distance formula I'm going to label u x sub 1 I'm going to label u y sub 1. I'm going to label u x sub 2 I'm going to label u y sub 2. now if you're looking at that and saying man that awfully looks like Pythagorean theorem but with points instead of lines and sides you're absolutely right it is the Pythagorean theorem so a lot of similar stuff is going to happen here where we Square things and maybe square root it maybe something nice happens maybe maybe not I don't know but this is the distance formula and if you look online and say oh I saw online that this is supposed to be x sub 1 minus y or x sub 2. it doesn't really matter because whether this is positive or negative once you square it it's going to be positive anyway who freaking cares so D is going to equal a big square root inside the first parentheses I have x sub 2 which I called negative 1. minus x sub 1 which I called 8. square that plus y sub 2 which is 3 minus y sub 1 which is 5 square that inside the square root I have negative 1 minus 8 which is negative 9 squared see what I mean this is just going to become positive anyway Plus 3 minus 5 is negative 2 squared okay let's bring that up here the square roots are getting smaller negative 9 times itself is positive 81 negative 2 times itself is going to be 4. and that gets me the square root of 85. I'm trying to think can I break that down any further no I'm trying to think like if I go and make a factor tree and try to rip it apart that gets me what Seventeen and five so that does mean no good so this is it I'm stuck I'm stuck at an ugly number now if you were if you had a teacher that said well what's what is that as a decimal it would be nine point something because I don't know what the square root of 85 is but I do know that the square root of 81 is 9. so something that's bigger than 85 is going to be something that's bigger than 9. so 9.2 blah blah blah but whatever your calculator says I'm not a calculator I'm a human I'm a human being and I have needs and I need you to smash that like button all three of you watching this thank you find the slope of each line the slope is the rise over run positive slope look like that negative slope look like that what is that you're straight up and down now any slope that's up and down is undefined up and down is undefined and left to right is zero do we see any left to rights here no but this is undefined why you ask well if I were to do rise over run and be like all right well let's pick the two points that they gave me fine up one two three four over don't move so four over zero what's four over zero undefined you can't do it now left to right would be zero because if I were to go like you know write five uh uh or up zero right five or whatever zero over five is legal you get zero but it's not what we're doing get get out of here this is a negative slope so negative my rise it doesn't matter which way you go from dot to dot so I'm going down one two three four so down four and I'm gonna go over one two three four ah negative four over four I wonder if I can simplify that I can negative one your slope doesn't have to be a fraction it can be but it doesn't have to be this is a positive slope my rise is one two three four five six so my rise is six my run is one two three so six over three is positive two so if you're like I hate the points that they gave us why could they just give us this point and I can go up two over one you can and I just did so stop complaining stop talking like that ridiculous but this is how you find slope graphically easy peasy lemon squeezy find the slope of each line well it's giving us lines in slope intercept form so if I remember my algebra 1 slope intercept form says that when Y is all by itself the number in front of X is my slope so when wait a minute there's no y here the heck well if I were to graph this x equals negative 1 would be a vertical line what's the slope of a vertical line on defined so the slope of this guy where there is no Y is undefined now over here Y is all by itself when Y is all by itself the number in front of X is my slope so the number in front of my X is 3 over 2 so my slope is 3 over 2 like that okay so that's how you find the slope if you're given something in slope intercept form or something that's awfully close to slope intercept form now if this was y equals negative one then this slope would be zero because it's a horizontal line but that's not what I asked you so stop it write the equations of five lines that are parallel to y equals x over 2 minus six parallel lines have the same slope but different y-intercepts okay so I'm going to view this from the Viewpoint of y equals MX plus b where Y is all by itself the number in front of X is my slope and the number floating around is my y-intercept now this is not written like that this is not a number in front of X this is X over 2. so why don't I rewrite this as instead of saying x divided by 2 I take half of X so I have half of x minus 6. so what I need to do is write out five of these guys that have one half in front of X and something that's not negative 6 floating around so how about y equals one half X believe it plus zero that's one how about y equals one half X Plus 68 there's two y equals one half x minus 61. oh tricky there see if I put minus six that doesn't count because it would be the same line and that's not parallel but minus 61. that's just fine how about y equals one half X plus four twenty three and how about y equals one half x minus six six seven oh the possibilities are endless but here's five of them and literally the possibilities are endless I can make a video where I could ask you 500 of these and you would just be so excited that you would watch the whole thing and I appreciate it if you did because that's what puts money in my pockets baby parallel lines classify each triangle by angles and sides if you want to classify a triangle by angles you're looking at three words if you see a 90 degree angle it's considered a right triangle if all of the angles are less than 90 it's acute like me and if one of the angles is greater than 90 it's obtuse now if you're like should it be all the angles are greater than 90 look if all three angles are greater than 90 you're going to have triangles with angles bigger than 180 that's not possible knock it off now if I wanted to name these based off of the sides that it has if all three sides are different you are looking at scalene okay all three sides are different if I give you a triangle where two of the sides are the same it's isosceles I'm always an afraid I'm going to spell that wrong but you know I'm a math teacher who cares if all three sides are the same like so it's equilateral for equal sides equilateral equilateral all right so those are my three things so let's take a look at what we have here this first triangle right here I have two sides that are the same which tells me that it's isosceles and one of my angles is a 90 degree angle so it's isosceles and right yeah there you go no numbers needed for this one for this one I'm going to look at my sides first and I see that side is different from that side which is different from that side so all the sides are different if all the sides are different that means we have a scalene triangle looking at all three angles 57 degrees 44 degrees 79 degrees all of these are less than 90 which tells me that it's a cute triangle and last but not least we have this triangle over here where this side is different from that side but oh these two sides are exactly the same so we have ourselves another eye sauce Colleen what makes this one different from this one is that angle is greater than 90 which means this triangle is obtuse so the types of triangle that we didn't hit we hit a right acute and obtuse but we didn't hit an equilateral and if you're thinking well equilateral and isosceles is an isosceles a type of equilateral vice versa there are many ways that you could name these but this is as specific as we can get okay so there you go find the measure of the missing angle all right so what do I have here I have myself a triangle right here the sides of these sides of the triangle just kind of keep on going forever what can I do what can I do well I do know that the insides of a triangle one two and we'll call you X are going to add up to 180. so let me do that 30 plus 20 plus I'm calling it X is going to equal 180. 30 plus 20 is 50. so 50 plus x equals 180 minus 50 minus 50. and that tells me that this angle right here x is 30 degrees or 130 degrees which is what I meant to say so let's get rid of X because X is just some arbitrary letter that I used so if x is 130 degrees how is that going to help me find this question mark well this question mark is vertical from 130 which means this question mark is the same as 130. so the missing angle is going to be 130. so a lot of times when you're asked to find missing angles and it's not as straightforward as you want it to be you have to kind of build like a puzzle until you get to what you want and what I wanted was found by doing the interior angle sum formula which is that guy finding out that it's 130 and so this question mark vertical from it is also going to be 130 so that's my answer find the measure of the missing angle all right so let's see what I have I have 155 an exterior angle I have 60 on the inside you know what I can do I'll have a ton of information but I do know that 155 is attached to what we'll call X and these two guys add up together to form a straight angle in other words they're supplementary so they add up to 180. so I'm calling you X just for funsies If I subtract 155 from both sides X ends up being 25. so let's erase I just found out that this guy is 25 degrees great now what I don't have is that question mark and I don't have enough information yet still to find out that question mark technically I do I'm not going to say that I do I'm going to pretend like I don't that's for a later thing but I'm going to pretend that I don't know something called the exterior angle stop and I'm just going to deal with the fact that you're 25 you're 60 and this missing angle here is going to add up to 180 so let me write that out 25 plus 60 plus let's call it y is 180 degrees that's the interior angle sum if I combine these that gives me 85 so 85 Plus y equals 180. If I subtract 85 from both sides 180 minus 85 is 95 so y equals 95. in this case degrees so let's erase U let's make U 95 degrees so this question mark is now completely adjacent and supplement to that 95 degree angle supplement is a fancy word for that 95 and that question mark is just going to add up to 180 degrees similar to what we did over here so if I subtract 95 from both sides subtract 95 from both sides that gets me 85 degrees now we may have noticed that 85 Degrees appeared in the green stuff over here and that's why I whispered X2 that's going to be in a future video but for this video I pretended that that didn't exist and I ended up with this question mark being 85 degrees foreign angle so the angle that I need to find is all the way the heck over here so usually when I have to do a wacky problem like this I'm dealing with the fact that I have triangles triangles everywhere that ends up dealing with 180 degrees I have angles that are glued together to make straight lines more 90 to 180 degrees I looked at that which is why I said 90 degrees we've got a lot going on here so let's kind of take what we have and start building pieces to a puzzle and work our way over here all right so I have 115 and this angle right here creating a straight angle so what I do know is that if I were to add 115 and that angle right there I'd get 180. so to find that I'm going to subtract it 115 from 180. I'm not showing all of my algebra steps I'm using Common Sense here and that gives me 65 degrees so you are 65 degrees now working my way from the left to the right I can find out what this angle is by using the interior angle sum in other words 65 plus 85 plus we'll call it X is going to equal 180. so let's combine 65 and 85 that's going to be 150 plus x equals 180 If I subtract 150 then X is 30 so X here is 30 degrees okay so still haven't gotten to my question mark but again I'm starting from the left and building my way to the right now this whole thing all three of these angles this little red one the right angle and that guy right there are going to make a straight angle 180 degrees so I'll call you y and I'll say that 30 plus this 90 degree angle right here plus Y is going to equal 180 degrees combine the 30 and the 90 and that's 120 so 120 plus y equals 180 subtract 120 from both sides subtract 120 from both sides and Y is going to equal 60. so this guy right here is 60 degrees Q still not at this question mark yet so what color didn't I use yet Purple I can find this angle right here by the fact that these two guys add up to make 180 in other words 180 minus 155 is going to give me 25 that chunk right there so I have 6C here I have 25 here oh I have another inside of a triangle that I can find so I know if I were to call you Z that 60 plus 25 plus Z is going to equal 180 interior angle sum once again 60 and 25 is 85 85 Plus Z equals 180 subtract 85 from both sides subtract 85 from both sides and that missing angle is going to be 95. so U are 95 degrees last but not least last but not least I'm going to do blue I'm going to end with the color that I started out with I'm finally able to figure out what my question mark is my question mark is going to create a straight angle with 95. in other words 180 minus 95 gets me my question mark and so 180 minus 95 is 85 degrees so it took a while but it's just like building a puzzle with the fact that we know that two lines that are created or two angles rather that create a straight angle is 180 and the interior angles and the triangle is also 180. so lots of 180s kind of like me changing my mind am I right because of the 180s like And subscribe find the measure of the missing angle missing angles way over here so what am I going to do I'm going to build off of the knowledge that I have over here essentially work my way left to right until I can answer that guy right there 75 is vertical to this guy making you 75. not so bad 45 is vertical to that guy making you 45 not that bad now no vertical here no vertical here but what I do have is this is the third angle in a triangle where I already have two of the angles so if I were to pretend that you're called X I can use the interior angle sum and do 45 plus 75 plus x equals 180. 45 plus 75 is 120 so 120 plus X is 180 if I were to subtract 120 from both sides 180 minus 120 is 60. so this x down here 60. 60 degrees all right that's 79 what can I do it's not attached to a triangle oh but you know what you know what 60 and 79 and this angle right here I'll call you y all of these are glued together to create a straight line which means 60 plus 79 plus y all adds up to 180 degrees so 60 plus 79 is 139 so plus y equals 180 subtract 139 subtract 139 and Y is going to equal 41. so you are 41 degrees now if you're 41 and you're 68 then this guy right here which I'll call Z is going to be the missing angle of a triangle which means 41 plus 68 plus Z all add up to 180 degrees interior angle sum so one so 41 plus 68 is 129 plus Z equals 180. subtract 129 subtract 129. and Z is going to equal 51. so Z let me erase that doesn't if you're wondering like why is he using x y z why those letters doesn't really matter in fact I don't have any of those answers in my answer I haven't even gotten to my answer yet I'm just using X Y and Z to help fill in blanks I haven't gotten to my answer yet until now because question mark and 51 are glued together to create a straight angle which means 51 plus question mark is going to add up to 180. so if I subtract 51 subtract 51 from both sides my question mark is going to be 129 degrees so there you go uh lots of puzzle building lots of going from left to right to figure this guy out but I figured it out because I'm a freaking genius find the measure of the missing angle all right we are now going to pretend that we know the exterior angle sum an exterior angle is an angle that is kind of like created from one of the lines of a triangle going a bit too far so I know that the exterior angle is going to be the sum of the opposite interior angles so whatever angle C is not attached to you add these up a plus b and that gets you my exterior angle why is that important here well because question mark is an exterior angle and question mark is going to be the same as 35 Plus 95. because those are my interior angles that are not attached to the question mark 35 Plus 95 is 130. so my question mark is 130 degrees and that's it I'm done find the measure of the missing angle all right what I have here is an exterior angle I know the exterior angle is equal to the sum of its opposite interior angles so that's what I'm going to set up I'm going to say my exterior angle 18x plus 5 is going to equal the sum 46 Plus negative 1 plus 8X looks a little sloppy maybe should have put parentheses but you know what too late too late I have 18x plus 5 on the left that hasn't changed yet on the right I have 46 plus negative 1 or 46 minus 1 which is 45 Plus 8 x let's get rid of the X because I have an x on the right side and on the left side so let's get rid of the one on the right side because I know that's going to make this a perfect 10x which is a sweet term 10x is a type of sugar look it up also called confectionary sugar or confectioners sugar I don't know I I don't do carbs so I have 10x plus 5 equals 45. minus 5. minus five cross UL 10 x equals 40. divide both sides by 10. and X equals 4. now the directions that I have say find the measure of the missing angle I made this problem and I think I just made a mistake so I'm not going to find that measure there you know what I am I created this mess I have to fix this so let's find out what this guy is right here let's find it out okay uh what do I know I know that this guy is 18x plus 5 and that if I were to add these two guys up they add up to make 180. so what I have is 18x but you know what X is four so 18 times 4 plus 5 plus this question mark is going to equal 180. 18 times 4 is 72 72 plus 5 plus I don't know equals 180 this makes 77 plus I don't know equals 180 and if I subtract 77 from both sides I'm out of room then the missing angle that I have is going to be 103. degrees now some of you are like why did you do it that way couldn't you say this plus this plus I don't know equals 180 and then plug in 4 for that yes I could but I didn't there's two ways that you could do it and once again I think that was a typo that I said find the missing the measure of the missing angle I'm pretty sure that was from our previous problem and I didn't delete it but I went with it because it was fun I had a great time I hope you did too the answer is 103 which is two more Dalmatians than my favorite movie about Dalmatians 101 Dalmatians subscribe to Disney plus right now find the measure of angle y d c y d c is this guy right here okay so I have to find out what this angle measure is is 15x plus 5. all right well I don't know what x is oh but you know what this is an exterior angle these are opposite interior angles and the exterior angle is always going to equal the sum of its opposite interior angles in other words 80 plus 6X plus 6. now I probably should have wrapped parentheses around 6X plus 6 but I know I won't need them so I didn't fight me the left side doesn't change 15x plus 5 doesn't change on the right side I can combine 80 and 6 to make 86 and then I add 6X to that because that hasn't changed now I have a variable on the left and a variable on the right let's get rid of 6X because y naught I could subtract 15x from both sides but this keeps things positive and I'm a cheery guy so I plan on doing that uh 15x minus 6X is 9x drop down the plus five drop down the equal sign drop down the 86. let's subtract 5 from both sides and I get myself a nice little 81. perfect because my next step is to divide both sides by nine which tells me that x equals nine hooray I'm done right no because my job is to find angle y d c angle y d c is given to us by 15x plus 5. it's not saying 5x it's saying find out what x is then plug it into this guy and the measure of angle y d c is going to equal that so 15 times 9 plus 5 is going to be I believe that's 135 plus 5 which is 140 degrees oh man cool beans so I did it I did it triangle b a c baby got that is congruent to Triangle L M N okay so here's pictures oh it wants me to figure out what angle a is congruent to two ways that you could do this angle a is telling me that it has two little hoopy things which matches up with that M over there which also has two little hoopy things which means angle A's can go into angle M the other way that you could figure this out is a is the second letter here and M is also the second letter here and the only way that you could say two triangles are congruent to each other is if you line them up the exact way that they're supposed to be lined up so I can look at this and say that c and N are congruent and B and L are congruent without looking at the pictures I can tell you that side AC is congruent to m n so on and so forth we're doing here I'm giving you two congruent triangles and I'm saying look a corresponds to which one over there m that's it I'm done write a statement that indicates that the triangles are congruent so when you're writing out a congruency statement you're going to say triangle blah is congruent to Triangle blah it's important that it doesn't matter which letter you start out with so if I start out with D I have to make sure that the order I put it in has the thing that corresponds to d d has three lines attached to it I guess that's an I has three lines attached to it so D and I correspond to each other so over here if we pick a different letter and it doesn't matter which letter that you pick it doesn't matter the order if it's just a triangle so if we go with c c has two hoopy things attached to it so does H so C and H correspond to each other last but not least I only have one letter left so I have B which matches up with G B and G and we can say that triangle DCB corresponds and is congruent to Triangle IHG in that order now the reality is we have an infinite amount of not an infinite like four or five ways that maybe six ways that we can write this out I can write this out as CBD is congruent to HGI or I could do B C D is congruent to ghi as long as D matches up with i c matches up with h m b matches up with g it's all good as long as you put things in the corresponding order foreign state if the two triangles are congruent if so how so we are going to look at two methods of triangle congruence we are either going to deal with SSS and SSS says if I have three sides of a triangle here that match up with three sides of a triangle over there then these guys are congruent SSS and I'm also going to look at SAS this is where you have to be a little bit more careful s-a-s says that if that side is congruent to that side that side is congruent to that side the angle in between the included angle has to be congruent too okay s a s where the angle in between so these are the two methods that we're going to look at so I see a side matching up with the side I see another side matching up with another side and you might be thinking um that's two sides and I don't see any hoopy things for angles neither of them incorrect don't forget about this included side right here homie aside is equal to itself that's reflexive property so this guy is SSS or if you're a snake it's my favorite geometry joke so if you're thinking oh okay I wonder if the same thing's going to happen here probably we have a side matching up with the side we have an angle matching up with an angle and the included side here side angle in between side side angle in between side this is SAS right over here we have a side matching up with the side we have an angle here matching up with an angle and we have these two sides are the same now before you get excited this right here side angle side this one is not this is angle side side also known as you know what not going to write it down but what I know is SAS is not the same as ass they're not the same so are these two congruent no these two were yes this is a no there are many ways that you could tell triangle congruence the curse word is not one of them not at all state what additional information is required in order to know that the triangles are congruent for the given reason so I need to prove that these two are congruent by SAS SAS is called s-a-s because you have a side and an angle in between and a side so if these two sides are congruent the angle that I have to draw must be in between the two sides the included angle similarly GH and IH have to have H as my angle so the information that you need is you need to show that angle L is congruent to angle h once I show that I can say side angle side side angle side the two triangles are congruent I'm done state if the two triangles are congruent if so how the two triangle congruencies we're going to look at is Asa and a a s now in a previous example we did SSS and what was it ASA no SAS okay we do an ASA here also ASA is a way that you can call your friend anyway what does ASA look like well if I give you two triangles ASA means I have to show that these two angles are congruent these two angles are congruent and most importantly the included side the side in between those angles have to be congruent too AAS says if I have two triangles I can have an angle and an angle an angle and an angle but a side that is not in between be congruent so the difference between ASA and AAS is the order of the letters where ASA says the side is in between the Angles and AAS says that the side is not in between the two angles so what do I have here I have these two angles are the same I have these two angles are the same but I don't have a side oh wait a minute yes I do it's the included side in between these sides are the same exact side which means they're equal so I have angle side angle a s a what do I have over here I have a side I have a side I have an angle I have an angle no other pictures I don't have an included side oh but you know what I have vertical angles so I have angle angle not included side angle angle not included side a a s all right I have a side and a side uh uh no angles nothing's attached to each other there's no right angle thing I can't even prove anything or these two angles or triangles congruent I don't know but do I have enough information to say if they are aren't nope so these two guys are congruent this guy is not because I only have two sides and nothing else and I continue to ask my snake friends is there a one and they said no it's a snake joke for you because of the two s's but you have to be careful because there's SSS which is but notice I didn't say I said because two s's are shorter than three please watch these videos and comment and like I'm trying as hard as I can here state what additional information is required in order to know that the triangles are congruent for the reason given all right two triangles I need to show that these guys are congruent by ASA ASA in this order says that an angle and an angle must have a side in between so angle angle I need that side in between angle angle I need that side in between so in order for these guys to be congruent I already have my two angles but I need to show that side KL is congruent to side I have to make sure I do it in order u t now a lot of you are like why didn't you put to you and why did you put UT why did you stutter for a second there because K matches up with you and T matches up with L I have to make sure I put it in order but now I have enough information to prove that these two angles are congruent based on ASA because I made the side in between those two angles congruent done state if the two triangles are congruent if so how right triangles right triangles right triangles I have four right triangle congruence theorems and those are HL h a l a l l n l a good start what does HL mean HL means hypotenuse leg if I give you two right triangles and I show you that one of the legs is congruent and the hypotenai I don't know are a congruent that's enough information for me to say these two triangles are congruent to each other h a says if I give you a right triangle and I have an angle and that matches up with a right triangle and an angle here and the hypotenai are congruent that's enough information for me to say that these two are congruent by ha it's a really funny theorem LL which is a really cool theorem LL Cool J the singer from the 90s he's a rapper says that if I have uh two right triangles and I can show that that leg is the same as that leg and that leg is the same as that leg I have enough information for me to say that these two guys are congruent and la which is a city in California if I have two right triangles and if I can show that one of my legs is congruent and one of my angles is congruent then that's enough information there so you might be looking at these and saying like isn't that the same as like SAS and isn't that the same as AAS yeah they pretty much are it's just that we are making this now towards right triangles and I'm not going to overthink it so right triangle angle hypotenuse are the same that's h a okay angles hypotenuses right triangle ha uh what do we have here we have a leg and a leg and a right triangle but I have nothing else do I I do these two legs are glued together so we have LL right triangle LL Bean isn't that a thing and then we uh uh h here H here L here L here right triangles right triangles HL so the only one that I didn't do was LA but you know you guys get the idea those are my right triangles state what additional information is required in order to know that the triangles are congruent for the given reason I need to show that these two triangles are related by LL Cool J which is a rapper from the 90s I made that joke in a previous video I'm not going to stop until I stop doing math for a living I need to show that these two are congruent well LL implies that these two guys are right triangles which they are good I have one leg and I have the other leg so how can I show that these two triangles are congruent using LL should I choose these lines no because that's an hypotenuse I'm going to choose these two lines so I need to prove that line C A that's my daughter coughing There She Goes Again is congruent to x v There She Goes Again now I'm putting it in that order C matches up with X and A matches up with v you have to make sure that things are in order when you're writing out congruent statements I did ca for cough and XV for but now I've proven that these two guys are congruent there you go find the value of x what do we have here well these two angles are the same oh if the angles are the same that means this is an isosceles triangle and these two opposite sides are the same these are called the base angles okay and when the base angles are the same that means their opposing sides are also going to be the same as each other long story short X is 6 undone now again isosceles triangles say that if these two angles are the same then these two sides are the same which tells me that X is six short and sweet find the value of x all right X is that guy right there oh this is equal to that which means that this is an isosceles triangle and the base angles are going to equal each other so you're 75 and you're thinking well that doesn't solve for x does it no but what I do know is if that angle 75 and that angle is 75 then X is the missing angle in a triangle I can use the interior angle sum and say 75 plus 75 plus x equals 180. 75 plus 75 is 150 so 150 plus x equals 180 subtract 150 from both sides subtract 150 from both sides cross U out and X is going to equal 30 or 30 degrees it's very tempting to look at this and have exterior angle thoughts which I guess is no not really possible you don't want to do that they just added that line to throw you off a little bit don't be fooled which I believe JLo once told you to do that don't be fooled by the rocks that I got and I'm here to tell you don't be fooled by the exterior angles that I've got I'm still I'm still Mr Parrish from the block like And subscribe find the value of x all right so I'm starting over here with the 120 and I'm ending up over here with an X I have these two sides are the same these two sides are the same they look like they could be equilateral triangles but what I know for a fact is this guy right here is isosceles so I'll label that you guys are the same as each other okay I'll deal with a similar Factor over here but I'm not there yet what do I have I have 120. oh this is an exterior angle an exterior angle is going to be the sum of the opposite interior angles inside the triangle now I'm not going to call this x because I'm supposed to find X I'll call you y so 120 is the same as y plus y so I'm going to set up an equation Y plus y the sum of the interior angles equals 120. or 2y equals 120. divide both sides by 2 and Y equals 60 degrees oh look at that and this is kind of nice because since those guys are 60 degrees that must mean that this third angle is also 60 degrees so we do have an equilateral triangle will that help us answer the question no just a fun fact now what can I do moving my way down here I know that these two guys are the same but what I also know is that 60 is vertical to this angle right here making that 60. if these two sides are the same then these two angles are the same making U 60 degrees making you 60 degrees which doesn't making U equilateral which doesn't matter and what does matter is the fact that X and 60 are vertical to each other which means X is 60 degrees so through the knowledge of isosceles and equilateral triangles we were able to find out that this whole thing just was a bunch of equilateral triangles glued together and getting US 60 degrees for x find the value of x all right the measure of angle two is 13x plus 3 there it is and I have 146 way the heck down here so I kind of have to make my way down to that sort of making my way downtown but making my way from 2 to 146 that's a different story you know what I'm going to do I'm going to make my way up I'm going to move on up to the east side so let's do this what I know is this creates a 180 degree angle okay this glue to that makes supplementary angles so I can say 180 minus 146 is 34 degrees so this guy right here let me clean that up is 34 degrees all right what do I have here I have myself an isosceles triangle which means this angle and this angle are exactly the same 34 degrees which means this is 34 degrees why because these guys are vertical and since this is also isosceles you're 34 degrees so after all of that what I know is angle 2 and 34 add up to make 180 degrees so 13x which is angle 2 plus 3 plus 34 add up to make 180 degrees because they're supplementary combine like terms that gets me 13x plus 37 equals 180 subtract 37 subtract 37. 13 x equals 143 divided by 13 divided by 13 and let's put it up here x equals 11. not degrees I shouldn't put degrees there I shouldn't because X is just some number and x equals 11. very tricky if you were to try to go from the top right to the bottom left it would be an absolute nightmare so I guess you know the what we want to do the strategy is start with the number and then make our way towards our X which in life don't we want to make our way away from our exes who's with me ex-girlfriends and boyfriends like And subscribe find KJ found it we're done just kidding all right this is a mid segment a mid-segment is a line in a triangle that connects a midpoint of one side to a midpoint of another side by definition the mid segment of a triangle is half of the base of the triangle in other words KJ is half of AC KJ is half of 18 KJ is nine so pretty simple mid segment Nothing to Lose a whole lot of Sleepover I did it solve for x all right X is just some arbitrary variable what I have is a triangle w r x o r is a mid point because these two are congruent and Q is also a midpoint because these two are congruent which means QR is a mid segment by definition a mid segment is one half the measure of the base so we could say q r is one half the measure of YW QR is given to us as 2x minus three that's going to equal one half of YW which is X plus nine and I'm going to put that in parentheses because you need it now it's just a an equation with X on both sides of the equal sign we do distributive property like so 2x minus 3 is untouched equals half of x plus 9 . uh let's get rid of half of an X so minus one half x minus one half x two x's minus a half is one and a half or I'm gonna write out three over two x drop down the minus three drop down the equal sign you're now gone drop down the positive nine now it's a two stepper so I'm going to add three to both sides add 3 to both sides two thirds x equals twelve and rather than dividing both sides by two-thirds let's multiply both sides by the reciprocal of two-thirds which is three over two so I'm going to multiply both sides by three over two that allows me to Cross these guys out and the trick when you multiply a whole number to a fraction is do 12 divided by the denominator so 12 divided by two is six and then six times three is eighteen so in this case all I had to do was find out x x ended up being eighteen uh the age of of being an adult here in the United States find the VW Volkswagen Happy Gilmore VW all right well W first off is a midpoint because these two are congruent V is also a midpoint because these two are congruent which tells me that this is a mid segment VW is a mid-segment by definition a mid-segment is half of the base of a triangle so VW is half of NL so VW was X Plus 15. that's going to equal one half of NL which is X Plus 21. now it's just an equation with distributive property X on both sides so let's distribute out the one-half that gives me X plus 15 that part hasn't changed half of X is one half X and half of 21 is 21 over 2. okay you could put 10.5 you could put 10 and a half I'm keeping it as 21 over 2. I have a feeling that's just going to be the best for us you'll see why now I have X on the left and half X on the right let's get rid of the half X on the right by subtracting one half X from both sides one half x one whole x minus one half X is one half X drop down the plus 15 drop down the equals you're gone drop down the 21 over 2. now what I'm going to do while I'm at it is I'm going to Quick replace this Twi with this 15 I almost said 21 I got 21 on the mind uh what was that 15 I'm going to turn 15 into 30 over 2. because my next step is to subtract 15 or 30 over 2 from both sides where did I get 30 over 2 because 15 divided by or multiplied by 2 over 2 is 30 over 2. and just a little bit of multiplication so I'm going to subtract 30 over 2 from both sides subtract 30 over 2 from both sides cross U out drop down the one-half x equals 21 over 2 minus 30 over 2 has me subtracting the numerator so negative 9 but keeping the denominator over 2. my final step is rather than divide both sides by half multiply both sides by its reciprocal so I'm going to multiply both sides by 2 over 1 which is the same as 2. that crosses these out X is now alone on the left negative 9 over 2 times 2 has me cross out the twos and leaves me with just the negative 9. now my job here is to find v w x is negative 9. oh the temptation to Circle X is negative 9 is there but that's not what I'm supposed to do now that I know that VW is X plus 15 and X is negative 9 I do negative 9 plus 15 which gives me 6. so v w is 6. and we're done and you might be thinking uh so then what's the point of MN there was no point of MN it's just there so thanks thanks for nothing MN thanks for nothing the figure shows a triangle with one of its angled bisectors find X if the measure of angle two equals 1 plus 28x and the measure of angle x v w is 59 x minus one all right this is angle two right there and that's 1 plus 28x v w is this whole angle here as I was very poorly Drew but there it is now an angle bisector sounds exactly what it is it takes an angle and cuts it in half so the measure of angle two and let me actually write that up here just in case I need the space I don't know if I do but I'm just going to do it up here the measure of angle two Cuts xvw in half so half of the measure of angle x v w so what's the measure of angle two one plus Twenty Eight X that's going to equal one half of the measure of angle xvw which is 59 x minus 1. now the other way I could set this up is I could have said that this guy equals two times the measure of angle two either way you'll get the same answer I've already committed so 1 plus 28 x hasn't changed I'm going to multiply one half to 59 over X so I'm going to put that as 59 over 2 x minus 1 times a half is minus a half uh what I'm going to do is I'm going to since my next step is to subtract 59 over 2. I'm going to turn 28 into something over 2. now if I'm going to turn 28 into something over 2 I multiply 28 by 2 which is 56. so I have 56 over 2 x let's drop down the one let's drop down the equals 59 over 2 x minus a half in order to get rid of this X over here I'm going to subtract 59 over 2 x again subtract 59 over 2 x cross U out 56 over 2 minus 59 over 2 is minus 3 over 2. drop down the one drop down the equal sign drop down the minus a half now what I have is a terrible terrible terrible two-step equation I've heard of terrible twos but terrible two-step equation that's not much worse at all I have a two-year-old than terrible twos are a nightmare I'm going to subtract one from both sides negative 1 over 2 minus one and let me write it over here negative 1 over 2 minus 1 is negative three over two so on the right I have negative 3 over 2 on the left I have negative 3 over 2 x that's actually pretty nice because what I could do is I can multiply both sides by its reciprocal which is negative two over three I know that looks sloppy but if I multiply both sides by negative 2 over 3 that completely cancels out which means that also completely cancels out and x equals one now my job is to find X x equals one I don't have to do much more than that it wasn't pleasant looking but that's what I got the end find PV if PW is seven now I had to add the little blinds here because I forgot to put the rest of the directions these are all angle bisectors you're an angle bisector you're an angle bisector you're an angle bisector if you have the three angle bisectors of a triangle drawn out they meet at a spot called the in center now the neat thing about the in Center is that the distance from the end Center to each side as long as the sides are perpendicular to that distance that's just how the definition Works anyway they have to be perpendicular from point to a side it's got to be perpendicular it's just a rule all of these guys are going to be congruent to each other so PW is 7 which means PV is also seven which means PX is also seven I didn't tell you that but that's just a rule this is sometimes called the in Center theorem but again by definition if you have your three angle bisectors drawn out they meet at the end Center and the in Center if you were to draw three perpendicular lines all three of those perpendicular lines are going to be the same measurement so in this case PV is seven because PW is seven and for those of you who live in southeast Pennsylvania perk Valley and Plymouth White Marsh PX nothing XP experience points like And subscribe the figure shows a triangle with one or more of its medians all right so what's a median a median connects an angle to the midpoint of its opposite side so technically this happens an angle to the midpoint it's of its opposite side technically that happens find LJ which is you if I J is six all right not so bad this whole side is six the whole side is six this is the midpoint which means LJ is going to be half of the entire side half of the entire side so LJ is going to be half of 6 which means LJ is three dunzo funzo find K if x t equals and KB equals one of the things that I forgot to mention in the directions is all of these are medians okay all of these are medians and what a median does is it takes the midpoint of one side and connects it to the opposite angle it doesn't mean that these are angle bisectors it just means that these guys are the same now is that what I have to do I have to find KT ah so this is even worse KT is this line right here uh and KB is this guy right here the whole thing now this is called a centroid doesn't really affect anything I just wanted to call it a centroid now what what you do with medians is whatever the angle or the vertex is to the centroid that's going to be two-thirds of the entire line okay so KT is two-thirds of the entire line KB okay so this is going to take a few minutes KT is 6x plus 6 all over five that's going to equal two-thirds of 11 fifths x minus six over five now what I'm going to do to make my life a little bit easier is I'm going to combine these two is it necessary no but I'm going to combine these two I have six x plus six over five nothing's changed there two-thirds nothing's changed there 11x minus 6 all over five eleven x minus six all over five like so now what I can do is I'm going to multiply 2 3 so 2 is going to be multiplied to everything there 6X plus 6 over 5 still unchanged that's going to equal 2 times 11 which is 22 the Taylor Swift special minus 2 times 6 which is 12 all over 3 times 5 which is 15. now what I'm going to do is I'm going to crisscross applesauce that's why I combine things in the first place I'm going to multiply 15 to 6X plus 6. 15 times 6 is 90. so that becomes 90x and I'll write it up here plus 90. I'm Distributing 15 to both of these 19 or 15 times 6 x is 90x 15 times 6 is 90 equals I'm going to crisscross applesauce again 5 times 22 is 110x 5 times negative 12 is negative 60. and now I have variables on both sides of the equal sign so why don't I subtract 110x from both minus 110 x minus 110x that gets me negative 20x plus 90 equals negative 60. let's subtract 90. let's subtract 90 and that gets me negative 20 x equals negative 150 divide both sides by a negative 20 divide both sides by a negative 20 and x equals positive seven and a half not fun at all that's medians or just not fun find the coordinates of the centroid the centroid is found by basically connecting the mid points to all of the angles no there's a lot of mid stuff going on here so it would make sense that what you do is you take all the X values so I will label them by calling them x sub 1 x sub 2 and x sub 3 you add all three of them up and divide by three it's basically finding the average do the same thing with the Y values y sub 1 y sub 2 y sub 3 add them up divide by 3. all right so let's find out our three x values uh that x value is negative five so negative 5 plus that x value is zero I think I can handle that plus that x value is 2. so negative five 0 2 divided by three my y values are zero this goes all the way up to six so plus six and this is zero again because that point is two zero divide U by three negative five plus zero plus two is going to be negative five plus two which is negative three over three zero plus six plus zero is six I think I can handle that six over three negative 3 divided by 3 is negative 1. 6 divided by three is two and that's my answer so negative 1 2 as you can see again it's found by connecting all of the angle I want to say angle bisexuals because that's not true all of the mid points to their opposite angles so midpoint opposite angle midpoint opposite angle right in the middle there is your centroid there you have it find the coordinates of the centroid all right the centroid is basically connected by taking a bunch of midpoints and connecting it to its opposite Angle now there's no picture and you might be thinking what the heck does that mean don't even worry about it what you do is you take the three x values so we if I were to label these x sub 1 y sub 1 x sub 2 y sub 2 very slope formula e but I have x sub 3 y sub 3. you find the centroid by adding all of the X values x sub 1 x sub 2 x sub 3 and divide it by three do the same thing with the y y values y sub 1 y sub 2 y sub 3 divided by three so that's going to be three plus negative five plus zero divided by three and that's going to be y sub 1 which is 4. plus zero plus zero divided by three three plus negative 5 is negative two plus zero is still negative two so this gives me negative two over three so I'll just write that out as negative two over three four plus zero plus zero just gives me four over three so I'll write it out like that and that's my guy and if you're like well there those are fractions that ain't good it ain't wrong no one likes fractions but there's nothing wrong with that answer so we're done foreign state of the three numbers can be the measures of the sides of a triangle this uses What's called the triangle inequality theorem and the triangle inequality theorem says that if you add any two sides of a triangle it has to be greater than that third side so if I were to label a triangle just any arbitrary triangle x y z x plus y has to be greater than Z X Plus Z has to be greater than y and what didn't I do y plus Z has to be greater than x so if I call you X Y and Z we could put all of those to the test X Plus Y is eight plus two has to be greater than eight yup checks out ten is greater than eight uh X Plus Z is going to be eight plus eight has to be greater than Y which is greater than 2 18 is greater than two that checks out and Y plus Z has to be greater than x so 2 plus 8 has to be greater than eight that checks out so are these the three sides of a triangle yay now's my daughter sneezing now we'll try it again this is X this is y this is z so I have to say three plus six has to be greater than two nine is greater than two check uh X Plus Z 3 plus 2 has to be greater than six no five is not greater than six so can these be the measures of a triangle no and that makes sense there's no way you can build a triangle that has a side of three a side of two which adds up to five and has it reach up to a six not possible so this one was a yes this one was a no and that's the triangle inequality theorem in a nutshell the two sides of a triangle have the following measures find the range of possible measures for the third side simple what you do is you take these numbers and you subtract the largest from the smallest that's three then you add these two that's 15 and your third side which we'll call X has to be in between them it can't be equal to it has to be in between so another way that we could say this is greater than three but smaller than 15. so this uses the triangle inequality theorem but uh really quick it basically says that look if if you needed to connect these you need something like 3.1 3.2 or you can have 14.9 you can't have anything bigger than 15 you can't have anything smaller than three so anything in between three and fifteen should do the job done do order the angles in each triangle from smallest to largest well this doesn't give us the angles oh but you know what it does it gives us the sides and whatever the largest side is in this case 18 its opposite angle is going to be the largest so let's find the smallest side first okay LK is the smallest side which means angle J is going to be the smallest angle because J is opposite of 14 which is the smallest side similarly the one in the middle is 16. so the opposite angle of 16 is K so angle K is going to be in the middle and then the largest angle is going to be L like I mentioned before because the largest side is 18 so the largest angle is going to be angle l so angle J angle K angle L because you were the smallest you were in the middle and you are the largest side and that's it order the sides of each triangle from shortest to longest well I don't have the sides I have the angles oh but you know what what I can do is I can find the shortest angle which is 55 and whatever the shortest angle is the side opposite that is going to be the shortest side so since my shortest angle is or my smallest angle is angle F the shortest side is going to be side g e so let's find the medium angle 62 barely but that's the medium angle the side opposite that is side g f so we'll say side GF for girlfriend and last but not least the biggest angle is 63 so the side opposite of that is going to be f e so my longest side is Fe so there you have it whatever the shortest angle is it's opposite side is the shortest side and so on and so on and we are done EC is the altitude not to be confused with Attitude which is something all three of my daughters have just kidding except I'm not uh altitude measure of angle one U is two x plus seven and the measure of angle two U is three X Plus thirteen find the measure of angle one and the measure of angle two all right what is an altitude an altitude is a fancy word for height now if you're going to measure the height of anything it has to be straight up and down so since e c is the height of this triangle right here EC has to be straight up and down and since this is a right angle of course that implies this is a right angle which implies that the measure of angle one plus the measure of angle two are in 90 degrees when added together they're complementary so that's how we're going to set this up we're going to say the measure of angle one plus the measure of angle two equals 90. so 2X plus 7 plus 3x plus 13 equals 90. all right we'll combine that 2X and that 3x because they are like terms becomes 5X another nice round number 7 and 13 make 20. and that equals 90. now it's a two-step equation subtract 20 subtract 20 5x equals 70. all right now it's a one step equation divide both sides by five and you get 14. done right wrong we're supposed to find the measure of angle one not just X and we know that the measure of angle one is two x plus seven so the measure of angle one is going to be 2 times 14. Plus 7. 2 times 14 is 28. and 28 plus 7 is 35 so the measure of angle one is 35 degrees so let me write that out a little bit cleaner the measure of angle one is 35 degrees now the measure of angle two is done very similarly okay up here measure of angle two is three X Plus thirteen so it's going to be 2 times not 2 now the measure of angle 2 is going to be done very similarly the measure of angle 2 according to the top up here is 3x plus 13. but you know what we know X is 14 just like it was up there in blue I don't know why I didn't just write it down I wasn't thinking I guess so 3 times 14 plus 13 3 times 14 is 42 42 plus 13 is 55 degrees so the measure of angle 2 is 55 degrees these are my two answers and this makes sense if I were to add these guys together 55 plus 35 I get 90 degrees which is what 1 plus 2 was supposed to be all right so that's altitudes for you fun fun all right find the coordinates of the orthocenter of a triangle with vertices 0 6 4 6 and 1 3. I know what you're thinking orthocenter isn't that where I go to get my braces tightened no that's the point where the three altitudes of each triangle intersect so let's draw a kind of nice picture of what we're dealing with here we have points at 0 6 zero six we have another point of at four six so four six and we have a point of one three so estimating Ball parking at one three all right so there is a picture there is a triangle and my job is to find out where my altitudes intersect now let's be smart here when you're finding out where lines intersect you're solving a system of equations we only need two lines we only need two altitudes so I'm going to make life easy and only find two equations of my altitudes now let's start with the easy one which is going to be that guy right there remember my altitude is a line that goes through a vertex and is perpendicular to the side that's opposite of that vertex so in order to find the equation of this red line and I'll call it red in order to find the equation of this red line I need a line that's perpendicular okay so it's perpendicular two zero six and four six and it has to go through one three okay there now what do we see well if it goes through 0 6 and 4 6 it's horizontal which means something perpendicular to it is going to be vertical and the equation of a vertical line is x equals a number well what number is it going to be well if it goes through 1 3 then it's going to go through my x value which is 1. so great I already have one of my equations already I hope finding another equation is just as friendly no not at all um I could pick any altitude I want let's pick you so it's we're going to call it purple because of colors and if you're colorblind then it's gray we need one that's perpendicular to these two points right here so perpendicular to zero six again and perpendicular to 1-3 this time this one is going to go through the point four six so I need to come up with a slope perpendicular to align connecting these two points so in order for me to do that I have to find the slope of that so remember you find the slope by doing three minus 6 the Y values 3 minus 6 over 1 minus zero three minus 6 is negative 3 1 minus zero is one so we have negative 3. now that is the slope of this line right here which looks right looks pretty accurate but [Music] so the slope that I want is perpendicular to negative three so something that's perpendicular to negative 3 is going to be the opposite reciprocal of negative three which means we take that 3 we make it one third and make the opposite of negative which is positive so the slope that I want is one-third so if I use point-slope form which I'm going to do y minus y sub 1 equals m times the quantity x minus x sub 1. I can come up with the equation of a line with a slope of one-third and now goes through the point four six so I'll label you x sub 1 y sub 1 you are my M let's plug it all in y minus 6 equals one third times the quantity X minus 4. now before you're all like all right well let's distribute an add six no need this is an equation and that's good enough for me now what I'm doing is I'm solving a system of equations or my two equations or this guy boxed in here and this guy boxed in here my x value is already solved for so life is not that bad right now and what I can do is I can rewrite this equation and just simply substitute one for X so y minus 6 equals one third one minus four y minus 6 equals 1 3 times negative three because one minus 4 is negative three y minus six one third times negative three is negative one add six to both sides and Y is seven nope Y is five I was gonna say seven wouldn't make sense that would fall out here five seems to be a whole lot nicer so since X is one and Y is five that tells me that my ortho center the point where all of my altitudes intersect is going to be the point one five so this problem on a frustrating difficulty scale was up there towards the ten not impossible I mean fortunately this was horizontal if it wasn't then maybe we would go up to a scale of 10 out of ten but I'll give it a nine just like me a solid 9. find the coordinates of the circumcenter of ABC with vertices a 0 3 B 0 negative one and C six negative one well let's just pretend that we have a nice little triangle here so I'm going to draw the triangle and there's two ways that you could describe what a circumcenter is if I and I'm not as you can see by my drawing I ain't good at this if I were to try to draw a circle connecting these three dots the circumcenter would be the complete center of a circle better yet the circumcenter is also where my three perpendicular bisectors intercept now a perpendicular bisector is you take a side like this guy right here you find the midpoint which appears to be you and you make a line perpendicular to it going that away you do it again find the midpoint which appears to be you and make a line perpendicular to it which goes that away and then we have these two lines intersecting at a point right there now I know my answer okay I know my answer it's three one there it is done but what I'm going to do is I'm going to go through the whole process anyway because in all honesty this is a very friendly triangle it's very friendly it's graphable I gave you a graph it's done I don't have to think too much but they're not always going to be this easy so what I need to do is I need to come up with so I'm just going to go through the motions knowing my answer already knowing what's going to happen what I need to do is I need to find the equation for this blue line right here and I need to find the equation for this red line right here find out where those two lines intersect it'll be three wet one and when I'm done I'm done so let's go through the motions okay blue I need the midpoint of these two points which is 0 3 and 0 negative one so I need the midpoint of 0 3 and 0 negative 1. and it has to be perpendicular to that as well it has to be perpendicular to that as well that's the equation that I need to come up with the midpoint of 0 3 and 0 negative 1 is the average of the X values so the midpoint is the average of the X values and the average of 0 and 0 is 0 plus 0 divided by two which is zero and the average of my y values is 3 plus negative one two divide that by two one my midpoint is 0 1. now to find the slope I could see it right here but let me find the slope the slope is going to be negative 1 minus 3 because I'm using the slope formula so negative 1 minus 3 is negative 4. over zero minus zero which is zero which is undefined so what I need is something perpendicular to that and something that's perpendicular to a vertical line has a slope of zero so y equals a slope of 0 X plus b which means my y-intercept and my y-intercept is one so the equation of this guy is y equals one all right so again I went through the motions to show you how this is done I'm not going to do that again okay the equation of my red line here is a vertical line and it goes through three so a vertical line is in the form x equals going through three is three so now I have my two equations where does the equation y equals or x equals three and Y equals one intersect how about at three one just like I said it would all right so these problems have a tendency to get more complicated this than this but there you have it and again if you can just imagine that Circle being drawn right here that dot right there is this Center that's what circumcenter means the perpendicular bisectors of ABC intersect at Point G are shown in blue all right so Yuri perpendicular bisector your perpendicular bisector you're a perpendicular bisector wait a minute which means you right here would be a circumcenter now if you don't know what that means right if I were to draw a triangle and attempt to draw a circle around it okay the perpendicular bisectors all meet at this circumcenter now why is this important well if my circle is going to hit all three vertices then the distance from that vertex to the circumcenter is the radius which means all three of these guys are the same which means all three of my red lines here are the same which means if a g is 9 BG is 9. p is the circumcenter of x y z p x is 4X plus 3 so where's PX or PX and p z is six x minus 11 find p y okay uh so if this is the circumcenter right try to imagine that I drew a circle connecting these three dots now there's no way I'd be able to fit this circle on this page or this slide or whatever you want to call it the circle would look something like this p is the center of that Circle so if p is the center of this invisible Circle that means p goes to x p goes to y p goes to Z all of those are a radius of the circle which means all of these red lines are the same so PX is the same as pz so if p x is the same as p z then 4X plus 3 is equal to 6X minus 11. now I have an equation with variables on both sides of the equal sign minus 4X minus 4X you become 3 equals two x minus 11. at 11 add 11 you become 14 equals 2x divide by 2 divide by 2 x equals seven Circle my answer no it's telling me to find p y I'm not given any information about p y none but you know what I am given information about PX I'm also given information about pz so if I just find out PX or pz find out what that is I get p y so let's do PX because the numbers look friendlier PX is going to equal 4 times x x is now 7. so 4 times 7 plus 3 which is 28 plus 3 which is 31. and if p x is 31 py is going to be 31. that's my guy triangle rst is shown below X U is negative 4y plus 20. x w is 6y plus 10. so X U is this guy oh my gosh that guy right there and x w is 6y plus 10 x w is where are you there you are there you is find the radius of the inscribed Circle okay well this is actually going to be nice an inscribed circle is created by connecting the dots that are created by these lines right here so if you were to try and I'm going to be so bad at this but I'm going to try my best to draw a circle connecting these three dots now that is an inscribed Circle it's a circle drawn inside of a triangle the nice thing is is this line ux or XU rather and xw they're all a radius of this circle which means they're all equal so X U is equal to x w so negative 4y plus 20 XU is going to equal 6x or 6y Plus 10. and now I have an equation with variables on both sides the equal sign add for y add 4y cross U out 20 equals 4X 4y and 6y I'm so used to doing X pardon me okay minus 10 minus 10. let me get out my calculator 20 minus 10 is 10 10 equals 10 y divide both sides by 10 divide both sides by 10 y equals 1. Circle it I'm done right no remember each one of these guys is considered a radius of the inscribed Circle so all I have to do is pick one of these and plug in one for y so let's pick x w because it seems friendlier x w equals 6 times 1 plus 10. 6 times 1 is 6 and 6 plus 10 is 16. so for this guy the radius of an inscribed circle is 16 units whatever those units are okay inscribed Circle your radius your radius your radius that's it done in the figure u v is a perpendicular bisector so your UV you're a perpendicular bisector it doesn't really look like it but it is of SW I see it and VW Volkswagen is an angle bisector oh VW is right here is an angle bisector of swt find X and Y so the picture is pretty much drawn accurately now it's just a matter of finding out what X and Y are so let's start with the X's because it's there in alphabetical order if you are a perpendicular bisector that means you and you are exactly the same and by you I don't mean you I mean I mean this line segment and this line segment are exactly the same so let's set it up 3x plus 1 equals X plus three I have an equation with variables on both sides of the equal sign so let's subtract X let's subtract X that gets me 2x plus 1 equals three subtract 1 from both sides 2X equals two divide by 2 divide by two x equals one that's one of my guys right there now I need to find y y it lives here and it lives here well which one should I use let's use the fact that I have 4y which is that symbol right there that angle right there your perpendicular bisector which means you're 90 degrees you're 50 degrees and your 4y so I could say 90 plus 50 plus 4y equals 180 degrees from the interior angle sum of a triangle combine like terms I'm giving you 140 plus 4y equals 180 subtract 140 from both sides 4y equals 40. divide both sides by four y equals 10. now I could use this information to answer a bunch of questions I could be like what's U.S and U.S would be like okay well X is one so us is three times one plus four it's four which means UW is also 4 which means SW is four plus four which is eight I could ask you find this angle then and you would be like all right well this angle is 40 minus y 40 minus 10 is 30. I could ask you what this angle is and you would be like well you're 90 degrees and if you're 30 and you're 90 then you're going to be 60 I could ask you all sorts of questions based off of this information I could say like all right well this angle is 4 times 10 which is 40 which means this angle is 40 and that's 90. so you're 50 which means you're 50 which means I could just go on forever not forever but for like five more minutes asking you questions about this all I had to do was find X and Y I gave you some bonus you're welcome translate this picture right here five units right and one unit up all right so this is a translation it's a transformation where all I do is take a shape and move it somewhere else I don't rotate it I don't flip it I don't reflect it it's just I take this shape and go so let's do this just because you might have a teacher that shows you this type of notation I can write this guy out as if I'm moving it five units right you take your X and you add five taking your x value and adding 5 means you're moving it to the right subtracting a number means you're moving it to the left since I'm moving it one unit up I'm gonna say y Plus 1. plus one means I'm moving up if it was minus a number I'd be moving it down was it necessary for me to do that no but if you have a teacher if your book is like instead of giving you words if it says translate according to this thing right here well that's what it means write five up one so let's move our shape and just to be extra sweet I'm going to list out the points when I'm done when you move a shape the best thing to do is you move it one point at a time so it doesn't matter which one you start out with I'll start out with t uh five right and one up so here's T right one two three four five go one up and when you label it you label it t Prime that little apostrophe tells you that we have taken an object and moved it the object before you move it is called a pre-image and the object after you move it is called the image all right so before I write out what all these are let's do the rest of these so I'm still going one at a time go right one two three four five up one put a dot there call it g Prime go to B go right one two three four five go up one put a dot there call it B Prime and then of course I'm going to do my best connecting these dots knowing very well that this is my greatest weakness and I'm doing all right baby now the shape didn't change it didn't get bigger it didn't get smaller it didn't flip it didn't rotate when I translate all I do is move it so here's my pre-image here's my image let's list out our new points in no particular order G Prime has me starting at the origin going right one two don't move up or down so right two don't move up or down is two zero T Prime and not to be confused with T Pain is I start at the origin go right one two three four don't move up or down so four zero and B Prime has me starting at the origin go right one two go down one two three four so right two down four is negative four and those are the points of my image so again translating says let's just move a shape don't do anything fancy when I'm done if you have to list out the points there they are pretty simple thank you describe the transformation so I have two shapes here a blue and a black or a black and a blue which is what you're going to be when I'm done with you I'm just kidding please don't notify me uh I have to figure out which one is the original is the black triangle of the original or is the blue triangle the original well if you look very closely the blue triangle has little apostrophes on them which means the original also known as the pre-image is the image in Black so I'm going from the black triangle to the blue triangle I'm going to give this in two answers one answer in words and one answer in notation form and if you're wondering why both do we have to do both nope but I don't know what your teacher is going to have you do so I'm going to make sure I have it written out in both ways so we pick a point doesn't matter which one we pick a point and see what happens I'll start at Z and in order to get from Z to Z Prime I go right one two down one so right two down one that's one way that you can write this out using words and many teachers would be like perfect good you're done or you might have to teach might have a teacher that says Nope write it out like this all right well right two is if you're going left or right you use your x value and right 2 is in the positive direction so we add two if it was left two we would minus two if you go up or down that's the Y value and down one is minus one up one would be plus one but since this is down this is minus one so again two answers that are exactly the same I wrote them both down so that you can see what they both look like just in case you have a teacher that forces you to do one instead of the other but that's how you describe a transformation and in this case this was a translation from one shape to its image rotate 90 degrees clockwise about the origin rotate that means we're spinning things around in this case we're spinning it around clockwise which goes like a regular clock goes like that when you spin something usually not all the time but usually you either go 90 180 or 270. 90 is one quarter of an entire circle so it's important that we know that 90 degrees is one-fourth of the spin now this is the way that I've been doing it for years and you might have a teacher that does it differently but this is what I do I'm pretending that I'm using a piece of paper and I'm physically spinning it this way one quarter of a turn so I take my paper and in this case I take this guy and I spin it one quarter of a turn click my answer is going to look like a shape that looks exactly like that that is my answer so what I'm going to do is I'm going to write out these points as they appear so that when I fix my page or picture I can draw those points out and I have a nice pretty little picture all right so let's find out what these points are in no particular order I can start out with I so I'm going to call it I Prime and I find out what I prime is as it appears and if I start at the origin it appears to be left one two up one two three so left two is negative two up three is positive 3 so I prime appears that it would be at left two up three negative two three do it again here's e so e Prime looks like I go from the origin left one two three up one two three so left three up three X appears to be left one two three four five so X appears to be left five don't move left five don't move so negative five zero and Q appears to be one two three four down one two so negative four negative two now this is what my answer is supposed to be like so if your teacher says take the original picture and rotate it 90 degrees clockwise you can't just spin it and say here teacher look this is what I ended up with no we spun it so that we could see what our answer will look like we write down the coordinates as they appear so that what I can do next is spin it back to its regular position and draw those points out all right so let's do that let's draw those points out I was negative 2 3 so I prime is going to be left one two up one two three put a dot call it I Prime e Prime has you going left three up three so left one two three up one two three that's e Prime X Prime has me going from the origin moving 5 to the left and then don't move at all so one two three four five don't move at all and Q Prime has me going left one two three four down one two right next to my original I so this was my pre-image and now what I'm doing is I'm connecting the dots to make my regular image and I'm always so bad at this let me let me try this one again so bad at that no not happy no not happy so if you had a oh my gosh not happy at all so if you had the the capacity to draw like a normal person unlike me you would realize that oh these shapes are exactly the same they didn't change at all no they didn't what they did is they rotated and this guy spun one quarter turn that way and looked like that guy so that's how I do rotations take your page spin it in the appropriate Direction write down the dots as they appear fix your page and uh draw off those dots connect them better than me and you're done rotate 90 degrees counterclockwise about the origin all right so when we take a picture and we rotate it we have to first care about our Direction counterclockwise is the opposite of clock clock go this way so counter clock go that way 90 degrees tells you how much you turn so the whole spin would be 360. so 90 out of 360 is a quarter of a turn 180 would be a half a turn in other words upside down and we don't care about the direction there 270 would be three quarters of a turn but here we have 90. so the method that I do is I take my picture or in your case you would take your page or the page that you're working on or the printout that you're working on and you literally spin your page in this Direction one quarter of a time and what that looks like is you take your paper and you lay it sideways in this specific Direction So my answer is supposed to look like this triangle quadrilaterally looking thing standing up I know my answer is supposed to look like this so what I'm going to do next is I'm going to write my points as they appear so it's good that I don't have numbers on here so the negatives and positives don't throw me off this a appears to be to the left and up so I'm going to write out my Prime as if I'm going to the left and then up so let's get specifics in order to get from the origin to a I go left one two three up one two three four so a prime is left three up four negative 3 positive 4. in order to get to D I go from my origin left one two three four five up one so D Prime was left five negative five up one positive one in order to get to M I go left one two three four five down one so M Prime is left five down one negative one and to get to W I start at my origin and I go left one two three four and go down one so w Prime is Left 4 down 1. now that I know what my picture is supposed to look like I can take my picture my page whatever it is that I'm working on and put it back to normal knowing very well that my answer is going to live somewhere around there so let's draw these points out I have a prime which was negative three four so a prime started the origin is left 3 up 4 there's my a prime so I'm starting to build my new image that's my pre-image here's my new image D Prime is left five up one so I'm going to go left one two three four five up one that was D Prime M Prime was negative 5 negative one in other words left five down one so M Prime is left five down one M Prime and W Prime I'm sorry W Prime is negative 4 negative one one two three four one W Prime and now of course the part that I'm awful awful awful at you connect the dots so you can see clearly that this is the exact same shape that you originally had just you know flipped around a bit that looks awful but what we have is this guy spun one quarter of a turn my shape if I could draw better doesn't change doesn't get bigger doesn't get smaller it just spins around spins a right round baby it's from a song but that's how you do rotations you find out the direction you take your paper you spin it however many times you're supposed to you write your points out as they appear you fix your paper you plot those points out connect the dots and then you can see you have the same exact shape just rotate it around somewhere that's how rotation is done describe the transformation all right so first things first I have to find out what the original picture is and what my new picture is well it looks like the blue guy here has apostrophes attached to them which means the black guy is my original also known as the pre-image now I didn't just take my shape and moved it I didn't just take my shape and reflect it so this is clearly a rotation of some kind now when I describe a rotation I have to do two things give the direction and how many spins it took so we'd pick one letter one letter and notice that K used to be left one up three and K is now right three up one which means I went in a clockwise Direction and since I only spun it a quarter of a turn I rotated clockwise 90 degrees now some of you might be saying whoa whoa whoa whoa whoa buddy what if I took k and when one quadrant over another quadrant over and a third quadrant over but I went counterclockwise yeah what if I did that and since it's three spins I wrote down 270. who's wrong now neither of us we're both right there's two possible answers here spinning 90 degrees clockwise is the exact same thing as spinning 270 counterclockwise so either answer is acceptable you just have to make sure that you get the right number down for the direction that you're going but that's it that's how you describe this rotation right here reflect across the y equals negative two axis all right usually I reflect over like the x axis or the y-axis but this has me building a brand new mirror y equals negative 2 means I'm going to have a horizontal line that goes through negative two so let's go to the y-axis since it's y equals let's go to the y-axis and go down to negative two there you are and build my mirror now what I'm going to do is reflect this guy up here the best way to do Reflections is now that we have our mirror drawn out you take each point each letter and find out how many spaces our guy is away from the mirror so Q for example is one two units away from the mirror which means Q Prime is going to appear one two units away from the mirror Q Prime all right I guess that's an X maybe I don't know maybe it's an L maybe it's a z i can't tell let's call it a z it looks like a z now that I look at it closely so you're a z Now Z appears to be one two three units away so one two three units away Z Prime I seems to be on the mirror so don't move it okay that's allowed to happen and E appears to be one unit away from the mirror so e Prime is going to be one unit away from the mirror and now the part that I'm always terrible at is drawing this guy out oh man why is it so far away that's not too bad it's not too bad either maybe I'm finally getting better at this so there you have it there it is there's our reflection you could see that this guy is completely reflected just get some bonus points and get some more time added to this YouTube video so I get a little bit more money we're gonna list what these four points out in no particular order I prime now I measure my Prime points from the original origin not from like this green line or anything like that so my origin is right there and in order to get to I prime I go left one two three down one two so I prime is left three negative three down to negative two e Prime was start at the origin go left one two three go down one so e Prime was left three negative three down one negative one Z is going to be start at the origin go nowhere left to right and up one so Z Prime is nowhere left or right up one so zero positive one and Q Prime has me starting at the origin and going right one two don't move up or down stays glued to the x axis so that's how you reflect across a specific axis in this case y equals negative two has me take the x axis shifting it down two but I still see my beautiful little refraction right there looks good I'm sticking with it foreign reflect across the y-axis ugh my y-axis is right here so what's going to happen with this picture is I'm going to reflect inside of a mirror so basically what's going to happen is whatever shows up over here is going to be opposite over here and whatever shows up on the left is going to be opposite over here so there's going to be a little bit of sloppiness but that's okay the y-axis is my mirror so I'm going to draw a line on the y-axis acting as my mirror I'm going to draw it all the way down even though it's not necessary all right there you go the best way to do a reflection is you take each point one at a time and find out how far away it is from the mirror so let's start with a because it's kind of like the clearest one that I can use a appears to be one two three four units away from the mirror so let's graph it one two three four units away in the opposite direction and call it a prime let's do it again with the other letters we're going to run into each other that's okay W is one unit away from the mirror so w Prime is also going to be one unit away from the mirror D is one unit away from the mirror so D Prime is also going to be one unit away from the mirror and M is one unit away from the mirror so M Prime is also going to be one unit away from the mirror so we take that we connect the dots the best that we can and as you can see we have the exact same image just kind of flipped not flipped uh reflected I like to view these as when my art teacher wanted to be lazy in grade school they would give us all pieces of paper and say I put a bunch of paint on the paper fold it in half and open it back up and you have a butterfly that's kind of like what we have here a butterfly or angel's wings whatever your religion is now what I'm going to do is I'm going to list what these points are uh just because you might have a teacher that asks you to so D Prime I'm going to go to my origin and the way I get to D Prime is I go left one up one two three four five so D Prime was left one negative one up five positive five a Prime has me starting from the origin and going left one two three four and up one two three so left four up three that's a prime I'm going to avoid a curse word and put W next I know it's spelled differently but you know you never know who's watching these things uh start of the origin go right one up one two three four so write one up four is W Prime and M I wish it was a g it could be dog I go right one up one two three four five uh uh write one up five and that's reflecting dog Dom um reflect across x equals y reflect across Y equals X this is hard for me to do visually but I'm going to try my best to do this okay I'm going to draw out the Y equals X line and as you can see I screwed that up already now it's not looking so bad this green perfectly straight line is supposed to act as a mirror so if you could take your head so this is your head and tilt it sideways just a little bit and pretend that it's lining up with that green line you are going to see that this shape is going to flip over it now it's hard for me to kind of line things up and find out where these points go I know from regular Reflections that I always count out how many dots I am away or how many units I'm away from the graph or from the mirror and I just graphed it on the other side but that's hard to see what I have here is Y equals X so this is what I'm going to do this is a trick y equals x and x equals y let me list out my four original pre-image points this might seem a bit too much in gratuitous but I guarantee you if there's anything I'm not it's gratuitous hmm L the original L is don't move go up two so the original L is zero two the original H is left two up one two three so the original H is left two up three the original Q is left two and up one so the original Q is negative two positive one and the original p is left one two three down one two so the original p is left three down to negative three negative two what I need to find is clearly not these guys but these guys the point of the image the reflection points how am I going to pull these off without using my graph well I'll tell you if you look at the original equation that we're given Y equals X this is telling us that Y equals X now and x equals y what does that mean these guys just flip 0 2 becomes 2-0 negative three I'm sorry negative 2 3 becomes 3 negative 2. negative 2 1 becomes one negative two and negative three negative two becomes negative 2 negative three all I did was flip the points I take these guys and graph them out L Prime let's go to my origin is right two don't go up or down L Prime H Prime is right one two three down one two put a dot there H Prime Q Prime is go right one go down one two Q Prime and P Prime is negative two negative three so go left one to go down one two three that's P Prime now for me I struggle mightily with graphing these so I'm going to notice that all right well H connects to Q so H Prime is going to connect to Q Prime Q connects to P so Q Prime uh connects to P Prime okay P connects to L so P Prime connects to L Prime sorry Q Prime and then last but not least L connects to H so L Prime to H Prime so again if you take your head and you tilt it about an eighth of a turn you could see the mirror image how this guy reflects to that guy that these guys if I knew how to draw would be the exact same shape and this guy is just a reflection of that okay not easy but the Y equals X trick is just that it's a trick and it's meant to make life a little bit easier if you know the trick and now you know it describe the transformation hmm uh all right well I have three types of Transformations I have a translation which means I just move things so T goes up and right U doesn't so it's not a translation a rotation has me rotate things but nothing's rotated it looks like it's just a mirror image because it is my job is to find out where the mirror would be well it's hard to see but the only way that you can get this black shape and this blue shape on top of each other and have it share a side is if the mirror was the side so my mirror is this guy right here and have that going on is going to look like this so my reflection my mirror is not the x-axis it's not the y-axis it's this red line right here what is the equation for that red line well y equals MX plus b where m is my slope B is my y-intercept it appears to me that my slope is up one over two up one over two and it's negative so my slope is negative one it also appears to me that my y-intercept is nothing so putting that all together I get y equals slope of negative 1 X plus nothing sloppy y equals negative X kind of weird but y equals negative X is my mirror it's the only way it works translate four units right and one unit down then reflect across the x-axis all right let's translate it first let's write out our points and then reflect it and write out our points again I'm supposed to go four to the right and one unit down so let's go one point at a time let's start with F f is going to be one two three four to the right one unit down F Prime go to the next point in no particular order G is going to be one two three four to the right one unit down call it g Prime and Y has me going one two three four to the right one unit down y Prime let's list out those points f Prime starting from the origin the way I get to F Prime is I go left one up one two three so left one is negative one up three is positive three Okay g Prime has me starting at the origin not moving left or right so that's zero and having me move up one so zero negative one is G Prime y Prime let me make that dot a little bit bigger you'll see why in a minute is I start at the origin go left one don't move up or down so y Prime is going to be left one don't move up or down so this is my first image this is my first transformation okay bada bing my second transformation now has me reflecting it across the x-axis so I'm going to draw my mirror that's why I made that that big I'm gonna draw my mirror across the x-axis now when you reflect an image you take a point find out how far away it is from the mirror and write it out that many units away on the other side so if I take G Prime for example and notice that g Prime is one unit away from the mirror I'm going to write out its image one unit away I'm going to call this g double Prime because it's my second image F Prime is one two three units away one two three units away is f Prime y Prime is living on the mirror which means y Prime and Y double Prime are exactly the same connect the dots connect the dots and you can see we have a reflection so let's write those points out uh F double Prime just to stay consistent if I start at the origin F double Prime appears to be left one down one two three so left one is negative one down one two three is negative three g double Prime appears to me I don't move but go down one so zero negative one F double Prime has me going I did I already did absolute prime y double Prime has me going left one don't move so negative one zero as you can see when I reflect over one of the axes not much changes except for the Y value in this case is now negative or the opposite I guess I should say this is a fun little shortcut if you memorize stuff like that but this is me doing two transformations in one coordinate plane awesome how many lines of symmetry does this shape have what is a line of symmetry it's a line that you can draw that cuts to shape exactly in half where the left side and the right side are exactly the same so when you have regular polygons like this you should be able to look at this and be like I should be able to cut this completely in half just like that you can so one way of cutting a regular shape into two equal sides where the left and the right are exactly the same as you go through the midpoint of opposite sides so we do it again we do it again and so far if I focus on the sides I have three lines of symmetry but 000 as Beyonce once said we're not done because if I focus on the vertices the corners I can also cut my hexagon in half like so through the vertices so if I look to the left side of that it's exactly the same as the right side of that so when I have regular shapes like this one I not only cut through the sides but I cut through the vertices as well so I can cut through you and I can cut through you and what I end up here with the vertices is I have three more so my total amount of lines of symmetry is three plus three let me get out my calculator carry the one divided by a billion six look at that it's a patriotic hexagon I love it State the most specific name for each figure all right we've got four of them here they're all quadrilaterals but I could probably be a little bit more specific for each of them for example we have a quadrilateral where two opposing sides are parallel and that's it that's all I've got this would be a trapezoid now there's different types of trapezoids there's a right trapezoid that's if you have a trapezoid that creates a right angle that doesn't happen here what's next well we've got one pair of parallel sides we've got two pairs of parallel sides this would be a parallelogram parallelogram okay this guy right here congruent congruent congruent congruent don't know anything about the angles don't know if they're parallel uh but what I do know is all four sides are congruent which means this would be a rhombus a wrong bus is what I took to school one time and went to a different place he he joke and then we have ourselves four right angles opposing sides are congruent and opposing sides of congruent this is a rectangle so a couple of the ones we didn't hit we didn't hit Square we didn't hit just a quadrilateral we didn't hit kite uh but you know I think those are pretty simple pretty straightforward those are quadrilaterals fun find the measure of each angle indicated all right so it looks like in this case I only have to find angle G now what I have here is a one two three four-sided figure also known as a quadrilateral the angles in a four-sided figure add up to 360 degrees I've been given three of those angles so if the angles add up to 360 Degrees then 65 plus 110 plus 87 plus I don't know I'll call it x equals 360 Degrees let's combine like terms one at a time 175 plus 87 plus x equals 360 175 plus 87 2 6 2 in that order or not in that order did I do that right that looks good to me plus x equals 360. in order to get X all by itself let's subtract 262 from both sides x equals 98 but we will say 98 Degrees okay 0.6 away from a from a pretty popular boy band from the uh late 90s or was it just 98 Degrees I think it was just 98 Degrees Nick Lachey pretty famous guy look him up solve for x all right what we have here is four sides so that whole thing is 360 degrees now there's two ways that you can do this uh but let's play dumb let's just say that all I know is I have four sides which means everything adds up to 360 degrees okay so 104 plus 10x minus four plus interesting 104 again plus 76 equals 360 degrees so if we combine like terms I have 104 minus four is a hundred plus a hundred and four is 204 plus 76 204 plus 76 is 280. and then I have plus 10x equals 360. two-step problems subtract 280 subtract 280 10x equals 80. divide by 10 divide by 10. and x equals 8 which is all I had to do now there's another way that you could do this problem let me show you that other way of doing this problem okay if these guys are the same right that implies that these guys are also going to be the same because what we have here actually you know what since I'm not given any markings I'm not definitively given that information although yeah it's got to be the case if these guys are the same then these guys have to be the same they have to be so I have 10x minus 4 equals 76. if I add 4 if I add 4 10x equals 80 divide by 10 divide by 10 and lookie there 10 9 8 76ers and two ways to do that problem both ways you'll get eight eight eight eight is great solve for x the figure is a parallelogram oh this is nice if the figure is a parallelogram we know a couple things these guys are parallel these guys are parallel does that help us here no what else do we know these guys are congruent and most importantly these guys are congruent which means we could say 2x plus 15 is equal to X Plus 15. subtract X from both sides because we have X on both sides of the equal sign two x's minus one X is just an X X plus 15 equals 15. subtract 15 subtract 15 you cancel out and coincidentally so do you so x equals zero and you might be thinking that can't be right you can't have zero distance well I'm not given the distance X is not the distance X is just some placeholder for these guys if I were to replace x with 0 you disappear and if I already Place x with zero you disappear and you both end up being 15. so 0 is the correct answer because the math uh ends up the way it's supposed to so that's how you do it solve for x the figure is a parallelogram all right what do we have we have two angles that are adjacent to each other and parallelograms if two angles are right next to each other those angles add up to 180 degrees now if they were opposite of each other they would be congruent but that's not what I have here what I have is two angles that are next to each other so they add up to 180 so 80 plus 11x minus 10 equals 180. combine like terms so 80 minus a hundred minus 10 is 70 so 70 plus 11x equals 180 subtract 70 from both sides subtract 70 from both sides because it's a two stepper 11x is equal to 110 in order to get X all by itself divide both sides by 11 divide both sides by 11 and x equals 10. so that's how you do it okay make sure in this case you set them up equal to 180 not set them equal to each other very easy to make that mistake we didn't make it so we got it right the figure is a parallelogram t e is four plus two x so what's t e your t e u r four plus two x e v is 4 x minus four now in a parallelogram the diagonals bisect each other in other words u e will be congruent to es but most importantly for us t e is congruent to e v so the way I set this up is 4X minus 4 because EV is congruent to t e so equals four plus two x so I have a variable on both sides of the equal sign so what I'll do is I'll subtract 2x from both sides minus 2X cross you out 4X minus 2x is 2x drop down the minus 4 drop down the equal sign drop down the regular 4. 2x minus 4 let's get rid of the minus 4 by adding 4 to both sides like so 2X equals eight divide by 2 and x equals 4. so Circle that and call it a day right wrong because it's not asking us to find X it's asking us to find t e t e is four plus two x x is four so t e is 4 plus 2 times 4. 2 times 4 is 8. like so and four plus eight is twelve and so t e tight end equals 12. well football reference for you football heads speaking of football head Hey Arnold remember that show like And subscribe solve for x the figure is a trapezoid all right what I have here is I have a median okay the median is basically exactly what it sounds like it's the middle part of a trapezoid or a parallelogram or whatever it is that you're dealing with whatever quadrilateral that you're dealing with so this is a trapezoid we have to parallel sides doesn't mean that they're parallel angles or parallel or they're congruent they're just parallel but this is the median which means it's directly in the middle the formula to find the median is one half times the sum of Base sub 1 plus base sub 2. so if we were to call u m and it doesn't matter which one your base sub 1 is your base sub 2 is so we'll call you base sub 1 base of two if I were to use this information to solve for x I would say that the median M Negative X Plus 21 equals one half of Base sub 1 which I called 17 plus base sub 2 which I called 11. so pemda says let's do everything inside the parentheses first so one half of 17 plus 11 is one half of 28. Negative X Plus 21 one half of 28 is 14. now I have a two-step equation let's subtract 21 from both sides minus 21. negative x equals negative seven and if negative x equals negative 7 then positive x equals positive 7. so there's our guy and if we wanted to check our work negative 7 plus 21 is 14. so this length would be 14 and 14 happens to be completely in between 11 and 17. it's the average what the median gives us right in the middle solve for x the figure is a trapezoid actually let's be a little bit more specific see how these guys are congruent this figure is an isosceles trapezoid and the rule for isosceles trapezoids is not only do they bisect each other but the diagonals are actually congruent if these weren't isosceles trapezoids they wouldn't be congruent but in this case these two e c and FD are congruent so since it gives us EC and FD I could set those guys equal to each other so EC is 20 FD is 5X minus 10. and I have a regular two-step equation okay 5x minus 10 let's add 10 to both sides add 10 to both sides cross you out 20 plus 10 is 30. drop down the equal sign drop down the 5X in order to get X all by itself we divide everything from five and let's flip it around I like my X on the left 30 divided by 5 is 6. so x equals six and if I wanted to check my work I could do 5 times 6 is 30 and 30 minus 10 is 20 just like EC is already so there you go isosceles trapezoid because these two guys are congruent which means the diagonals are also congruent the figure is a trapezoid uh specifically this is an isosceles trapezoid I should have put those markings there but when you have isosceles trapezoids there's some news angles next to each other that are not on the same side or I guess these would be considered bases so angles that are on different bases that are next to each other add up to 180 degrees so what I can do is I could say 6X minus 22 this angle right here plus 8X plus 34 angle R both of those add up to 180 degrees so I can combine like terms 6X 8X 14x minus 22 and positive 34 is 12. that's going to equal 180 and I have myself a two-step equation okay let's get rid of this 12 first by subtracting 12 from both sides subtract 12 from both sides 4X equals 168 14x not 4X but would that change things divide everything by 14 divide everything by 14 and X is going to equal 12. Circle that and call it a day right wrong it's not asking for x it's asking for the measure of angle R and the measure of angle r is 8x plus 34. X is 12 so the measure of angle R is 8 times 12 Plus 34. 8 times 12 is 96 96 plus 34 is 130 so the measure of angle r is 130 degrees now if you're like I don't believe you then what you could do is you can plug in 12 to X up here and get 6 times 12 which is 72 72 minus 22 is 50. so if this is 50 and that's 130 then those guys add up to 180 degrees which is what I originally said it would okay so that's how you add adjacent angles of a trapezoid if they're on different bases good morning find x what we have here is we have a rhombus couple ways that we can approach this uh but what I see is it gives me a rhombus that's cut in half right here and by cutting a rhombus in half you are creating an isosceles triangle why does that matter well because if you are 8X minus three then you are also eight x minus three you're 106 and these are three angles of a triangle so the three angles of a triangle add up to 180 degrees so 8X minus 3 plus another 8X minus three plus 106 equals 180. now if you're wondering well couldn't I have just done two times parentheses minus three sure but you know I set it up this way and I'm not stopping I'm not erasing a thing so I have 8X 8X that's 16x we're combining like terms I have negative three another negative 3 which is negative 6 and 106 which is a perfect 100. that equals 180. now I have a two-step equation first subtract 100 from both sides I think I can handle the math here 180 minus 100 is 80. 16x is on the left still in order to get X all by itself we divide 16 from both sides and 80 divided by 16 is 5. we're being asked to find X nothing more so X is five and if you wanted to double check your work by plugging in 5 for x and you get 40 minus 3 and 40 minus 3 and then all that up you get 180 but I'm gonna I'm gonna leave it like that okay thumbs up find X all right what do we have here we have oh a rhombus this side is congruent to that side which means these two guys are equal to each other so let's set that up eight equals X plus one a one-step equation minus one minus one and x equals seven very straightforward very simple problem that's how it's done find the measure of r s t all right where is rst rst okay uh what I have here is I have a rhombus cut in half now what I know is when you take a rhombus and cut in half you get yourself an isosceles triangle right here why does that matter because if your 5x minus 5 then you are also 5x minus 5 because isosceles triangles have the same exact base angle right here so I can set up an equation where this angle plus this angle plus this angle equals 180 degrees so let's do that 5X minus five plus five x minus five plus five x plus 10 equals 180. good news negative 5 negative 5 and positive 10 all cancel each other out negative 5 plus negative 5 is negative 10 negative 10 plus 10 is 0. nothing 5x plus 5x plus 5x is 15x and now I have a one step equation all I have to do is divide both sides by 15. and x equals 12. done nope because what I have to do is find the measure of angle rst what I know is this guy is 5X minus five x is 12. so that guy is 5 times 12. minus five which is 60 minus five which is 55 degrees so this guy is 55 degrees now since this is a rhombus and this diagonal cut this guy in half that means both of these are 55 degrees and in order to find the whole thing I take 55 and add it to itself meaning that the measure of angle r s t which was that whole thing is 55 plus 55 which is 110 degrees so this one required you to think outside the box a little bit or should I say think outside the rhombus [Laughter] [Applause] yes sir find the missing measure what do we have we have these two guys are congruent these two guys are congruent oh adjacent sides are congruent this is a kite you can imagine putting a line here and a line here and pretending there's a string and a couple bows and you have yourself a kite now what's important about this is these two guys are What's called the vertex angles which means F and H are the non-vertex angles so why do we care about the non-vertex angles well because they're congruent so if my job is to find out what f is I know H is the same so if I call you X then I can call you X now how does that help me right now it does it but what I do know is this is also a quadrilateral and quadrilaterals add up to 360 degrees so 66 plus X plus 70 plus X is 360. so let's set that up 66 plus X plus 70 plus X is 360 degrees combine like terms 66 plus 77 is 136 X Plus X is two of them so 2X and now I have myself a regular old two-step equation if we subtract 136 from both sides minus 136 2X is going to equal 224. divide both sides by 2 and x equals 112. now we want to find out the measure of angle F we called the measure of angle f x so the measure of angle f is X which means it's 112 degrees all right so again we add ourselves a kite these two guys the non-congruent angles are going to be congruent they're called the non-vertex angles and so we just set it up equal to 360 and got it thumbs up M E equals 20. me is 20. I wish [Music] what do I know about this shape oh these two guys are the same and these two guys are the same we have a kite the diagonals of a kite bisect each other which means you guys are the same doesn't help me here you guys are the same that does help me here because it says that m e is 20. so if m e is 20 that means e k is 20. and MK Mortal Kombat is found by adding 20 plus 20. according to this calculator that I have right over here MK is 40. find x what do I have I have these two guys are congruent and these two guys are congruent we have a kite now what information am I given I'm given 50 degrees right here and 8x plus 2 and these guys are created by taking CE which is technically a diagonal of this kite what do we know about the diagonals of kites well the diagonals of kites bisect the angle that it goes through and if things bisect angles that means this angle and this angle are exactly the same measure so I can say 8x plus 2 equals 50 because they're the same so now I just start out with the two-step equation minus 2 minus two drop down the 8x 8X equals 48 in order to get X all by itself divide both sides by eight divide both sides by eight and X is six and that's all we had to do here was just find X I don't have to plug it into anything if I wanted to check my work and do 8 times 6 which is 48 48 plus 2 is 50 looks like I did it right but yeah the diagonals of kite's bisect the angle pretty nice find the measure of angle Z all right there's Z so what I have is I have the fact that 7x minus 4 5x plus 20 they oppose each other and I have a hundred and five so this problem is not near as simple as I wish it was this is what I know these two sides are congruent these two sides are congruent this is a kite because the sides are next to each other not opposing each other now when you have a kite okay the angles in between the non-congruent sides are called non-vertex angles and these angles are equal to each other so this is what I'm going to do I'm going to find X by setting these two equal to each other once I have X that allows me to find out the measure of both of these angles does that arrive me at my answer yet no my job is to find that angle so once I find out this angle this angle will be the same I have a hundred and five and I would have three angles in a quadrilateral and the four angles in a quadrilateral add up to 360. so I have to start out by setting these guys equal to each other first finding out what these are and then I can finally take care of Z so 7X minus four equals 5X Plus 20. I have X on both sides of the equal sign so let's subtract 5x from both sides so subtract 5x from both sides 7x minus 5x is 2X drop down the minus four drop down the equal sign drop down the twenty I can add 4 to both sides like that 2X equals 24. divide both sides by two and x equals 12. again that's helpful because now that I have X I can find out both of these angles now they're both going to be the same so it's not like I have to plug in 12 into both of these and solve them separately no if I just find the easier one like this guy and do 5 times 12 right 5 times 12 plus 20 that gets me 60 plus 20 this is 80. so you are now 80 and you are also 80 because you two guys were congruent finally I could figure out what Z is so I will actually call this Z and what I know is that this is a quadrilateral and this angle plus this angle plus this angle plus this angle is going to equal 360 degrees so 80 plus 80 plus 105 plus Z is 360 degrees if I combine like terms 80 and 80 is 160 160 and 105 is 265. 265 plus Z equals 360. subtract 265 from both sides subtract 265 from both sides Z equals 95. so your 95 which means after all of this I can finally say that the measure of angle Z is 95 degrees so this one required a little bit of extra thought but these are the types of problems that I love you have to kind of look at what you have figure out what you have to do and sometimes it's not as simple as doing one mathematical process in this case we had to do three but it was great and it was fun find the area of this triangle right here looking at this triangle it doesn't really look like it's as obvious as it is but if I were to take this triangle and spin it around and make it look like this where now I could say that this whole thing is nine yards and this is 6.4 yards it's a little more obvious what I have the area of a triangle is one half of Base times height now the height is always going to be a perpendicular line going straight up and down to a base here I don't really see the height because you were like oh the heart should be up from here to here but it doesn't matter any way you spin a triangle you can get your height so by taking my triangle and kind of laying it flat so that the nine yards is flat on the ground I can see the 6.4 is going up and down so you're my base and you are my height so I could say that my area is one half of my base which is nine times my height which is 6.4 okay one half of 9 times 6.4 is 28.8 now before I Circle that and be very happy we're talking about the area here okay the area is always given in units squared so I'm going to make sure I say square yards for this guy right here okay depending on your teacher you might have a teacher that's like oh I only care about the number I really don't care about this part but you really should say that the area is whatever unit squared feet squared yard squared square inches square meters whatever it is okay so that's how you find the area of a triangle sometimes you have to spin it around in your mind which I did find the area of this trapezoid right here all right the area of a trapezoid is given to us by the formula one half times the height parentheses base sub 1 plus base sub 2. this trapezoid is kind of lying flat already and it gives us the height right there and it's five which one's base one which one's base two doesn't matter doesn't matter at all okay so the area of this guy is going to be one half of 5 times base sub 1 which I called 6 Plus base sub 2 which I called 3. all right so one half of five times nine this is probably one that I can do without the use of a calculator one half of 45 is going to be 22 and a half yeah now before you Circle that and call it a day it's asking for area and whenever you're being asked for the area you should always include the units but Square them okay so this is Miles so we're going to say 22.5 square miles all right don't ever just say 22.5 don't say 22.5 miles it's 22.5 square units okay so we're done trapezoid write the name of each polygon now polygons get their names by how many sides they have so let's count the sides and name them one side two side three side four side five side six side seven sides is seven-sided figure is called a heptagon okay one two three four five six seven eight nine ten is a decagon ten Deca gone now I'm marking the side so I don't accidentally over count so these little marks doesn't mean necessarily that they're the same I mean in this case they look like they are I'll put little check marks now one two three four five six seven eight nine is a nonagon and last but not least this weird looking creature at the end here is one two three four five six sides hexagon so when you do problems like these when you're naming things and you just ask to name the polygon you don't care if it's regular which means all the sides are exactly the same or not unless they want you to be specific technically this is a regular heptagon because all the sirens are the same I technically wasn't told that but it's a little obvious this is not regular these are not regular but again you know all we had to do was name them and so I did state if each polygon is concave or convex the difference between a concave or a convex is if you have all of the angles where the interior angles are less than 180 degrees it's convex now I know something is concave when it looks like part of the outside caves in so this guy is concave does this cave in a little bit yep concave does this cave in at all nope convex that's the difference between the two that's it state if each polygon is regular or not a polygon is regular if all the sides are congruent and all of the interior angles are congruent so you look like you are all congruent and you look like you're all congruent so I would say yeah that's regular you no I don't even have to mark anything that's obviously not a regular shape okay so regular not regular dunzo find the interior angle sum the interior angle sum is a fancy way of saying if I were to add up all of these angles how many angles live inside the whole thing well there's a formula for the interior angle Sun s equals 180 times the number of sides minus two n here stands for the number of sides so what I do is I count up how many sides I have I substitute n with that number do a little bit of PEMDAS and I have my answer so you are one two three four five sides so n is going to be five so using the formula for this guy I have S equals 180 times 5 minus two in other words 180 times 3 is 540. so the angles of a pentagon which is what I have add up to 540 degrees how many sides do I have here one two three four five six seven eight nine ten eleven twelve so n equals 12 in this case so the sum of all the angles is going to be a hundred and eighty times 12 minus 2. so the sum is going to equal 180 times 10 which is 1 800 Degrees total okay so there you have it interior angle sum of a polygon pretty simple formula to use dunzo find the measure of one interior angle we are given a regular one two three four five six seven octagon so we are given a regular octagon where all of the angles are exactly the same so this is what we have to do first in order to find one of the angle measures I have to find the angle measures of this whole thing which requires us to use the interior angle sum formula that guy right there 180 times the number of sides minus two so s is going to equal 180 times 8 total sides minus two 180 times 6 is 1080. like the snowboarding game for Nintendo 64. now this is the degrees of everything if all of these add up to 180 degrees that means one two three four five six eight of the same exact measurement eight of the same angle so I will say 8A for angle equals 10 80 degrees divide both sides by eight and you end up with 135. which means each interior angle is a hundred and thirty five degrees the only reason why we were able to do with this method is because this is a regular polygon if it's not a regular polygon that's require a little bit of extra thought but since this was regular it made life a whole lot easier so we're done find the measure of one exterior angle all right there's a rule that says the exterior angles and if you're like what's an exterior angle if you were to take each side and keep going with the side the exterior angle is the measure there so if I were to do this all the way around all the way around it's asking me what's the measure of one of these for every single convex polygon for every single convex polygon the exterior angle sum is always 360 degrees so what I have to do here is I have to find out how many angles there are so the way I do that is find out how many sides there are one two three four five six seven eight nine so there's nine sides which means there's nine exterior angles so I will call an exterior angle X for exterior and since there's nine exterior angles and they all add up 360. I'll say 9x equals 360. divide both sides by nine and x equals 40. so each exterior angle is 40 degrees and we can kind of see that I know we're not supposed to look at a picture and estimate but we could see that if I were to put a bunch of these they would each be 40 and if I were to glue them together I would get 360 degrees so that's my guy find the area of the regular polygon all right so the area of a regular polygon is found by multiplying one half times the perimeter times the apothem now if you're thinking is an apothem like a Vermin that lives outside at night and is really gross looking no that's a possum we're being asked to find a possum so I I like that joke that's a good joke uh what is an apothem an apothem is basically a side bisector that's going to be perpendicular to that side and hits the center so this right here is my apothem and what I have is one of the side lengths one of the side lengths is 12. all right well in order to find the perimeter I have the hypothemus 14. 0.5 that's the apothe in order to find the perimeter I have the fact that year 12 there's one side two sides three four five six seven eight so the perimeter is going to be 8 times 12. so the area is found by multiplying one half to eight times twelve times 14.5 all right so I'm going to go off to the side here and pretend like I'm not going to plug this into a calculator and I did this all in my head and I ended up getting 69 6. now do I have any units no if I had units I would say square units so I'm just going to put square units where are you anyway I'm allowed to do that because it's area but that's how you find the uh area of a regular polygon so yeah fun fun apothem find the area of the regular polygon well what do I have I have a one two three four five six-sided shape okay I know that the area of a regular polygon is one half times the perimeter times the apothem now this is the apothem so I know that you're going to be 5 root 3 Gross uh perimeter I don't know what the perimeter is but I do know that there are six sides and each side is ten so the perimeter is going to be six times ten and the area is going to be half times that so I should be able to do this without a calculator one-half times six is three so I have three times ten times five Roots three 3 times 10 is 30. so 30 times 5 root 3 is going to be 150 root 3. I don't have any units so I could say unit squared all right so that's how you do it uh root three two can't forget root three well that's a close one whoo solve the proportion a proportion is when you have one fraction equaling another fraction and when you see this it's a very simple method called crisscross applesauce you take any diagonals and multiply them together so 4 times R is four R set it equal to the other diagonals multiplied together 7 times 5 is 35. now I have a one step equation divide both sides by four and looks like you're going to get a decimal here you can write it out as a fraction but you know what the decimal is not that bad when you divide by 4 32 divided by 4 is 8 so 32 and 3 more is going to be three quarters or 0.75 however you want to write it out but that's how you do proportions crisscross applesauce solve the proportion a proportion is when you have one fraction equaling another fraction when you have that the method is very simple crisscross applesauce take any two diagonals and multiply them together now since down here you have B plus nine I'm going to write it out so that I have B plus nine in a parenthesis so I remember to distribute okay after you do one set of diagonals you put down that equal sign and then you multiply the other set of diagonals so six times B plus five and again I'm going to make sure I put B plus 5 in parentheses so this is a little more involved than a previous problem that I have a previous video the video right before this where I have to do distributive property twice and I'm going to have B on both sides of the equal sign so 4 times B is 4B 4 times 9 is 36. that's going to equal 6 times B which is 6B and 6 times 5 which is 30. now I have variables on both sides of the equal sign so why don't I do this why don't I subtract 4B from both sides because I like to keep my variable positive it's not necessary that you do it this way I'm just doing it okay try and stop me that gives me 36 equals 2 B plus 30. uh B is not by itself I have 30 in the way and I have 2 in the way so I'm going to subtract 30 from both sides first okay cross that out 36 minus 30 I'm going to write that up here 36 minus 30 is 6 and then now 2B is all alone on the right side divide both sides by 2 divide both sides by 2 6 divided by 2 is 3 so I'll move that over here so I can move the B to the left and then the end B is 3. and if you wanted to check your work you could do 3 plus 9 is 12 and 6 over 12 is a half cool and 3 plus 5 is 8 and 4 over 8 is a half cool and a half is a half cool and I'm done state if the polygons are similar so I have two pairs that I'm going to do in order to make sure that both polygons are similar you have to do two things you have to make sure that all of the sides are proportional state if the polygons are similar so I have to check if these two are similar and I have to check if these two are similar it's two things that you're looking for here you have to make sure that all of your sides are proportional that means if I have a quadrilateral I have to make sure that one fraction equals another fraction equals another fraction equals another fraction okay and I do that with corresponding sides so I would do like 10 seems to match up with 15 so 10 over 15 and so on and so forth and I have to make sure that all of my angles are equal okay so the angles have to be equal the sides have to be proportional so let's do the proportional part first I'm going to match up corresponding sides and I'll do small to big all right I'll do small to big so I'm going to see if 10 matches up with 15 so 10 over 15 and I'll go clockwise here see if that matches up with 14 over 21 9 plus 10. see if that matches up with 10 over 15 again and I have to see if that matches up with 14 over 21. now notice a lot of these are repeats so I don't have to overthink that but now I simplify fractions 10 over 15 is 2 over 3. 14 over 21 is 2 over 3. 10 over 15 again is 2 over 3. 14 over 21 is 2 over 3. so the proportional part checks out now I have to make sure corresponding angles are exactly the same too so 90 matches up with 90 and 90 matches up with 90 and 90 matches up with 90 and 90 matches up with 90. so the angles are good the proportional is good these two are similar okay now let's do this again over here on the right side let's start out with the proportional sides first so let's do small to big again so we'll start at the top and work my way around counter a regular clockwise 7 over 21 has to match up with 5 over 15. which has to match up over 7 over 21 again so I'm not going to rewrite that which has to match up with 5 over 15. so I'm not going to rewrite that if you're figuring this out if you see repeats just be lazy 7 over 21 is a third cool 5 over 15 is also a third cool we're in good shape there proportional yeppers now I have to make sure the angles match up now I'm not given a ton of information about the angles except that is 40. this corner is going to be 40 and up here that corner is going to be 130. but what it tells us is these guys are parallelograms so whatever this angle is the angle right next to it is going to be supplementary to it in other words 130 and this angle have to add up to 180. which means you would be 50. which doesn't match up with 40 which means no these guys are not similar so kind of tricky kind of tricky because it looks promising with the proportional sides part but then the angles are like nah I'm Gonna Get You I'm Gonna Get You the polygons in each pair are a similar find this scale factor of this smaller figure to the larger figure all right so the scale factor is basically a ratio of small to big this is the smaller figure this is the larger fit figure I put B for big not L for large don't know why I did that but too late Biggie Smalls the rapper um so we have to pick sides that are absolutely corresponding so since I have 18 and 18 and 30 and 30 then I can choose 18 from the small and uh 30 from the big because this 18 lives in between 18 and 21 this 30 lives in between 30 and 35 so that 18 matches up with that 30. so 18 over 30 can be simplified if I divide everything by three or can I divide by six I can divide everything by six if I divide the top and the Bottom by 6 that gets me 3 over 5. so some people might allow you to put the scale factor as a fraction I think usually in Geometry it's given to you as the word three to five or depending on a different book maybe it might give it to you as three to five it really all depends either way we're doing small to big so three to five if it was big to small then it would be five to three but again depending on whatever book you use all of these work because it's all telling the same story three to five the polygons are similar find the missing side length whenever you're being asked to find the missing side length of similar figures you're going to want to use proportions it doesn't matter which way you do it I always like to make my proportions big to small and then what you do is you make a proportion where you do a corresponding side over corresponding side equals corresponding side over corresponding side so if I look at this 12 matches up with the six on the small so I'm going to say 12 over 6. that's going to equal I don't know on the big one so I'll put an X over 5 on the small one now what we have here is a proportion a fraction equals a fraction is a proportion and the way you do a proportion is you find a diagonal and you multiply that this is called crisscross applesauce so I could do 12 times 5 which is 60. then you put the equal sign there and then the other diagonal 6 times x is 6x divide both sides by 6 x equals 10. so the missing side length over here is going to be 10 which makes sense because 5 over 6 10 over 12 10 over 12 simplifies to 5 over 6 so everything seems to work out on the math side of it which I'd hope so the polygons are similar find the missing side length now in a previous problem I was only given two pieces of information for example I was only given like that info so aside and I don't know and a side in the side now I'm given extra so that complicate things absolutely not what I like to do is when I'm given two similar figures and I'm being asked to find the missing side length I like to make a proportion of the big shape to the small shape so big over small I pick a side on the big shape that I know I have the corresponding part two so let's use what looks like a hypotenuse even though I know it's not I'm going to choose 53 or 63 on the big which matches up with 54 on the small so 63 over 54. now if you're like well do I do the same thing with uh 56 or 48 nope I don't need that don't need them at all just extra information I care about what that question mark is and I'll just call it X for now 35 over here matches up with I don't know over there so 35 on the big is going to match up with X on the small now I have a proportion and when you do a proportion you do crisscross applesauce so 54 uh times 35 is of course 1 890. I did that all in my head and I didn't pause the video so I can look it up now I put an equal sign there and I do the other diagonal 63 times x to 3x in order to get X all by itself you divide by 63 on both sides of course and I absolutely know that this is of course 30. so 30 equals x x equals 30 but I'm not even going to circle X because X isn't even part of my original problem the figures are assuming and find the missing side length the missing side length is 30. okay that's how it's done the polygons are similar find X now my job is to find that x value right here I'm given two similar figures I'll call you big shape I'll call you small shape whenever I'm being asked to do any kind of solving with similar figures I like to create a proportion with the big small matching or the big side big triangle it'd be nice if I knew how to talk matching up with the small triangle of course I said it's big small because I have rap music on the mind now I definitely definitely definitely care about this x right here so I need to make sure I match 63 up with that guy so on the big I have 63 and on the small I have 8X minus 2. I'm going to set that equal to let me see here I have 49 and I have 49 but I can't use this 49 because I'm not giving any information down there so that extra 49 is useless so I'm going to choose this 49 on the big over that 42 on the small because they match up now I have a proportion and when you have a proportion you're going to do crisscross applesauce pick any diagonal and multiply them together 63 times 42 of course is of course is 2646. no calculator needed here just my mind now over here is going to be slightly different I have 49 times two things and so whenever you see that you have to do distributed property and group that 8X minus 2. so my next step is do nothing here but distribute that 49 to 8. Now 50 times 8 would be what 400 so 49 times 8 would be 392. I'm just gonna sneeze quick about you now that was correct and so don't forget at The X 49 times 2 negative 2 I mean is negative 98. I have a two-step equation now so I'm going to add 98 to both sides add 98 to both sides and cross you out 2744 that's going to equal 392 X I've heard of p92x but 392x that's different divide both sides by 392 divide both sides by 392 and I think it's nine it's not it's seven I was just joking with you guys cross that out uh X is going to equal seven so I'll put that right here x equals seven yeah so there you have it a little bit of extra information there to throw you off a little bit of extra information here to make you do distributed property but you know similar figures finding missing sides is all pretty much the same you just set up a proportion uh using what you have don't do too much crisscross applesauce do a little bit of math and then you're done so fun Zo the polygons are similar find X now I'm not given a ton of information except for the fact that the scale factor from A to B is five to six so I'm going to compare a to B which I know is five to six on a I have 3x plus 11. so I'm going to match that up with 3x plus 11 and on B I have 42. so this is no different than a regular old find the missing side of these two similar triangles if we were given information in fact if I wanted to I could have made U6 and I could have made you or made u5 and made you six and the problem would end up being the same anyway all right so crisscross applesauce whenever I have a proportion you always multiply diagonals make an equal sign and then multiply the other diagonal so we'll do 5 times 42 first which is 210. yeah that looks right to me we're going to set that equal to 6 times 3x plus 11. oh in parentheses whenever you have a group and you do proportions you're going to have to do distributive property don't forget so 210 doesn't change 6 times 3x is 18x and 6 times 11 is 66. now I have a two-step equation subtract 66 from both sides subtract 66 from both sides cross U out and you're looking at 144. that equals 18x now it's a two one stepper divide both sides by 18 divide both sides by 18. and X is going to end up equaling eight so there you have it uh I believe this is my fourth find the missing side of a similar triangle type problem all of them are the same you just set a proportion crisscross applesauce do some distributed properties sometimes and then take it from there that's it the polygons are similar find X now this is what we have here and it's not as obvious as what we have in previous problems we're so used to seeing something that looks like this and something that looks like this and then you match things up crisscross applesauce we have here except it inside of this triangle right there and if I wanted to I could rip this apart and write down as much information as I need based off of what I just did that red triangle was 36 right here 27 right there and negative 4 plus 4 x right there that big triangle was 64 all the way across and if you're like what about that part I don't care about that part it was 72 and 96. and now what I have is two separate triangles it's you know that used to be one big triangle with a little triangle inside of it now whenever I'm finding similar triangles or missing side of similar triangles I like to compare the big to the small and make a proportion we have extra unnecessary information first off I definitely have to find X so I'm going to make sure if I'm going to compare the big over the small that I first start out by comparing 64 over negative 4 plus 4X definitely have to do that first now if you're wondering all right so what other sides do I do doesn't really matter I could do 96 over 36 or 72 over 27 it doesn't matter so I'll do 96 over 36 because it doesn't matter at all now I have a proportion and whenever you have a proportion you do crisscross applesauce so we are going to multiply negative 4 plus 4X times 96 and when you do that you have to make sure you set up a distributed property problem because this is two separate things being added together negative 4 plus 4X then you have an equal sign and you multiply these two numbers together 64 times 36 is 2304 I did that in my mind now I distribute 96 times negative 4 is going to be 384 negative 384 plus 3 um 184 x equals 2304. if I add 384 to both sides that gets me 384 x equals 2688 divide both sides by 384 and I think it's eight but let me go off to the side here and double check glad I did because it's actually seven whoo but that's how you do it a triangle within a triangle just view it as two separate triangles rip it apart if you have to which I did and you're good to go the polygons are similar find the missing side length the scale factor from A to B is five to six and I have I don't know but it matches up with 30. so whenever I'm being asked to find the missing side length of similar polygons whether it's triangles or in this case a quadrilateral the rules don't change it's asking me to do a to B so I'm going to set a proportion which is a fraction equaling a fraction giving me this information A to B is five two six that's going to be equal to a is I don't know so I'll call you X over matches up with 30. now when you have a proportion you do crisscross applesauce which means you multiply diagonal numbers doesn't matter which one you start out with 6 times x is 6X then you drop down the equal sign and multiply the other diagonals 5 times 30 is 150. now I have a very simple two-step equation just getting one step equation and my one step is to divide both sides by 6 and 150 divided by 6 is 25. so there you have it there's my X I really shouldn't Circle X because X wasn't in the original problem I really should say just 25 I don't think you'd have a teacher that would Mark you wrong but if we really wanted to be specific we would just say the missing side is 25 and there you have state if the triangles are similar if so State how you know they are similar and fill in the blank what do they mean by state how you know well there's three ways that you can prove two triangles are similar there's SSS there's s-a-s and there's a a now SSS means is there a way that you can show that that side over that side so Like A over B equals that side over that side so C over D which also equals that side over that side which is e over F now if you're wondering what's like what are these letters this just means like that side over that side is the same as that side over that side which is the same as that side over that side a c and e are on the same triangle b d and f are on the same triangle so that's what SSS means it's not the same as the SSS for triangle congruence you're not looking for exact numbers you're looking for a ratio s a s means you're only going to pick two sides and match those up but you're also going to see if two angles are congruent a a is if two angles on one triangle are congruent and another pair of angles are a congruent that means automatically the third angles are congruent but we don't care about the third angle because all I need is two so what do we have here well I'm not giving any information about angles which means I can't use either of these two so I have to go with SSS so what I need to do is I'm going to compare big to small I always compare big to small when I can okay the longest side is 84 which matches up with 14 on the smaller one so if I'm doing big to small 84 is the longest side that is going to go over 14. equals now I'm going to do the same thing again I'm going to choose the middle one 72 which matches up with the middle one over here 12 so 72 over 12 and I'm going to do with the third side because SSS means you have to do it with all three 48 over here matches up with 8 over there 48 over 8. now I simplify each fraction and see what I have 60 or 84 divided by 14 is 6. 72 divided by 12 is 6. 48 divided by 8 is 6. since all of these fractions equal these are similar and they're similar using s s so state if the triangles are similar yes if so State how you know SSS and fill in the blank c b a is similar to I have to make sure that I'm good here C matches up with the F over here how do I know that C is in between red and black f is in between red and black C matches up with F B is in between green and black G is in between green and black so G and then the only ones after h so these were similar SSS was the reason and triangle CBA is similar to Triangle fgh that little squiggly means similar different from congruent state if the triangles are similar if so State how you know they are similar and fill in the blank if I want to see if things are similar there's three ways that I can do that I have SSS which means uh something on the big side needs to match up with something on the small side and the other thing on the big side needs to match up with another thing on the small side and another thing on the big triangle needs to match up with something on the small triangle that's SSS I also could use s-a-s which is are two sides proportional and our two angles congruent if so SAS or I could use a a which says are these two angles the same are these two angles the same oh well then the third angles are going to be the same a a now this is what I'm given I'm given two sides and that's it here's the other thing though I'm not given this but I know that that those are vertex angles which means these are the same so what I can use is s a s but I can't just be like it looks like SAS I'm done no I have to check my work okay now m v and 28 match up I always like to do big to small so 28 on the big has to match up with 14 on the small very good that has to match up with 49 over 8. hmm now 28 over 14 28 over 14 is 2. 49 over 8 is definitely not two so are these similar are they no they're not similar if so blah blah blah blah blah blah blah blah blah these are not similar therefore I'm done they're not similar because even though SAS looks promising the proportions are not the same and this makes sense I mean if you just look at the picture I mean no way not even possible I mean these two just like no way find x what I have is a triangle that is a right triangle and I have two other right triangles living inside this now there's a way that you could set this up and solve it using proportions but I'm going to show you a different trick called the geometric mean you may have heard of the arithmetic mean this is the geometric mean if you are a given a big right triangle and you have the altitude and your job is to solve for the altitude and that's what I have here if I were to take this triangle that's a hundred uh whatever is long and lay it flat okay like that I can see that X is my altitude what I know is one of those lengths is 36. now let's take that out pretend like we don't know that let's let's pretend we'll know any of this stuff all right get rid of these numbers okay there is a little shortcut using the geometric mean if you are trying to find the altitude of a right triangle what you do is you label part of this hypotenuse a and the other part B the altitude splits it and you set up a proportion a over x equals x over B that's the geometric mean now I have something very very very similar I have one of these guys is 36. I showed that over here I don't have the other guy but what do I know I can find the other guy by doing 100 minus 36. which is 64. so I would let A or B it doesn't really matter because you're just going to end up doing crisscross applesauce anyway I'll say 36 over X equals x over 64. now that I have a proportion I can do crisscross applesauce multiply two diagonals together x squared and multiply these diagonals together that gets you 2 304. take this square root of both sides because that's how you undo a square now usually usually you attach a plus or minus to your number here however I'm not going to do that because that implies you're taking a negative side length which doesn't make sense what I do get is the square root of x squared is X and the square root of 2304 is 48. I knew that right off the top of my head didn't use a calculator didn't use a calculator at all but this is using geometric mean there's other ways that you could do this by using similar I mean if I wanted to I could rip this apart and pretend that I have a small triangle and a big triangle and do similar triangles like I've done a thousand times on this channel but here I'm doing geometric mean to show you something new find x what I have here is I have two parallel lines and pieces of that parallel line there are several ways that I can set this up but one way that I could set this up is I have a big triangle with a small triangle living inside of it my big triangle is this whole thing my small triangle is this little tiny piece right here and so I'm going to set up a proportion comparing the big triangle to the small triangle so on the big triangle I have 45 that matches up with 5x on the small triangle okay on the big triangle over here where I'm given some information I have 36 on the big that matches up with 20 on the small now that I have a proportion I could do crisscross applesauce take two diagonals and multiply them together 45 multiplied to 20 is 900. five x times 36 5 times 36 is 180. so 180x divide both sides by 180 and x equals 5. so this is not too different from regular roll find the missing side of similar triangles problems that we've done before it just gives you information about parallel lines and parallel lines you mean you can do some special stuff with it but that's it we're done find x what we have here is we have three parallel lines and two lines that are not parallel that cut through it now there's a little Rule and I'll try to draw this similar to what we have here it's obviously not going to be the same but there's a little rule that says let's call you a let's call you B let's call you C let's call you D if these three lines are parallel then what we could do is set up a proportion A over B equals c over d now we have something very similar but it's incomplete see I of 25 here I don't have that chunk so 25 would be C I don't have a oh but you know what I could do I have to do 35 this whole thing minus 25 which is 10. okay here I have 22 but I don't have this part right here but I do have the whole thing now if what I did to find this chunk over here was do the whole thing minus 25 then why don't I do the whole thing minus 22 in other words 7 plus 14 x minus 22 which is 14x combined like terms negative 15. so I'll call you 14x minus 15. now I can set up a over B 10 over 22 equals c over D 25 over 14x minus 15. so let me erase this chunk right here and do just that 10 over 22 equals 25 over 14x minus 15. we have a proportion so you could do crisscross applesauce 22 times 25 would be 550 we're going to set that equal to 10 times oh a group so I have to set up distributive property 10 times 14x minus 15. so my first step in this problem is to do distributive properties so 550 just drops down [Music] 10 times 14x is 140x 10 times negative 15 is minus 150 now we have a two-step equation I can add 150 to both sides when I do that I get 700 equals 140x one more step divide 140 from both sides and X is 5. now there are several ways that you could do this there are several proportions that you could use but I'm using the rule that we're given and that rule is you have three parallel lines that over that equals that over that I could also have done 10 over 25 equals 22 over 14x minus 15. it'll all work out anyway because crisscross applesauce gives you a little bit of maneuverability and that's it we're done find X there it is aha this is a right triangle and we are being asked to find the missing side of a right triangle whenever you're being asked to find the missing side of a right triangle use a little old guy called Pythagorean theorem a squared plus b squared equals c squared now what's what well to be honest it doesn't matter which one A or B is as long as your longest side is c c is the hypotenuse C is always going to be the side not attached to your right angle so C has to be 13 in this case which one has to be a or b it doesn't matter as long as the longest side is C so I'll call a12 and I'll call BX now you substitute these numbers and that variable in here and do a little bit of solving a is 12 so 12 squared plus b is X so x squared equals c is 13 so 13 squared now the next thing that you do in Pythagorean theorem is you simplify the numbers so 12 squared is 114 plus x squared is still x squared and 13 squared is 169. in order to get X all by itself I have to subtract 144 from both sides so minus 144 minus 144 cross you out drop down the positive x squared equals and 169 minus 144 is 25. now squared is in the way I have to find X not X Squared how do I undo a squared square root both sides now you may have learned in the past that whenever you take the square root of both sides to solve you have to attach one of these guys a positive or negative symbol to the number but we don't do that here why because we're finding the missing side of a triangle so you're not going to say my answer is positive or negative well you can't have a negative side so it can't be or a negative So my answer in this case is just positive 5. now since I'm given a unit of measure I'm going to make sure I attach that in my answer so find x x is 5 inches and that's my answer we're done find X we are being asked to find the missing side of a right triangle whenever I'm asked to find the missing side of a right triangle we use this fella the Pythagorean theorem invented by a Pythagoras way back in the day now a squared B squared and c squared they are all representative of three sides here now what letter gets what well C has to be the hypotenuse the longest side the hypotenuse is always going to be the guy not attached to your right angle which one's a which one's B it doesn't matter as long as the longest side is C so we'll call A4 we'll call B3 and we'll call c x now I plug them into the formula a I called 4 squared a fun game to play B I called three so three squared and C is X x squared whenever you're doing the Pythagorean theorem whether it's you're finding a missing hypotenuse or a missing leg the second step is always to simplify these numbers 4 squared is 16. plus 3 squared is 9 still equals x squared if you can combine like terms combine them 16 plus 9 is 25 25 equals x squared now I didn't find x yet I have x squared and in order to get rid of that squared I have to square root both sides now in the past you were taught that if you take the square root of both sides in order to solve you attach a positive or negative symbol to the number we don't do that here because we can have a positive or negative side length you can't have negative side lengths it's only going to be positive so we're not going to attach that symbol so I'm sorry I even brought it up but the square root of 25 is 5 the square root of x squared makes it just X and what I'm going to do is I'm going to flip it around because I like my variable on the left x equals 5 and I'm going to make sure I attach the unit of measure which is miles in this case because it's proper but you are asked to find the missing hypotenuse and in this case finding the missing hypotenuse is Pythagorean theorem combine like terms square root you get five miles I would walk 500 of those to answer another problem it's a song look it up state if each triangle is a right triangle this uses a little thing called the converse of the Pythagorean theorem the Pythagorean theorem is a squared plus b squared equals c squared usually we use this guy to find missing sides of right triangles well in all three of these cases I'm given all three sides so what I have to do is use this guy to make sure that these are right triangles now what does that look like I go to one of my triangles and I label the three sides a b and c but oh do be careful because whenever you label a b and c you must call the longest side C because the longest side would be the hypotenuse in this case it doesn't matter which one a is it does matter which one B is and what you do now that we have things labeled is you plug it into this form formula so a squared is 12 squared plus B squared is 9 squared equals c squared is 15 squared and you might be thinking what are we solving for nothing the goal of using the converse of the Pythagorean theorem is to make sure our math works out if it does it's a right triangle because it works for the Pythagorean theorem if it doesn't it's not a right triangle it's still a triangle but it's not a right triangle so let's do the math 12 squared is 144. 9 squared is 81. 15 squared is 225. 144 plus 81 is 225 does 225 equal 225 last time I checked yes yes it does so this first one is a right triangle okay let's try it again and again so I have to label a b and c which one has to be a b or c well A and B doesn't matter as long as you call the longest side C so I'll call you a and I'll call you B so 10 squared plus 49.5 squared has to equal 50.5 squared 10 squared is 100. 49.5 squared is let me do the math my head 2450.5 and 50.5 squared is 2 550 uh 0.25 I meant to put a 2 here I don't know if I said it and just typed it or wrote it out wrong but there you go so um okay 100 plus 2 400 is 2 500. uh uh that equals 2 550.25 so since the both of them equal each other even though my I have a little writing errors here uh I fixed it and yes it does work out okay so not as pleasant um but you know I did the math in my head I did I did I did I didn't uh last one now I don't know the square root of 115 is but I do know that the square root of 121 which is 11 uh is 11. so this has to be C here because 11 is going to be bigger than the square root of 115 because 121 is 11. so I'll call you a I'll call you B and I'll set it up and you might be thinking oh no square roots don't worry about it it's actually not bad I have 9 squared plus the square root of 115 squared has to equal 11 squared 9 squared is 81. 115 squared cancels each other out if I square rooted Happy Days 11 squares 121. 81 plus 115 is like 196 or something which is certainly not 121. so nope nope all right so this is how you use the converse of the Pythagorean theorem I give you three sides I ask you is this a right triangle you determine which one has to be C and the longest one's always going to be C you plug it in you make sure it works and if it does it's a right triangle if it does if it's a right triangle if it doesn't it's not a right triangle not bad state if the triangle is acute like me obtuse or right we know that we could use the Pythagorean theorem Converse to determine if something is a right triangle by saying a squared plus b squared equals c squared that's how you know if something's right but how do you know if something is acute or obtuse well if you take the sum of the two smaller sides and that's what A and B represent Square them if that sum is greater than c squared you have yourself an acute triangle similarly if I were to take the smaller sides and square them and add them together and this sum of those is smaller than c squared you have yourself an obtuse triangle so I have to look at these and I'm going to say that you're my a and you're my B and if you're like whoa whoa whoa how did you figure that out A and B doesn't matter as long as you call the longest side C so I'm going to say let me get a color I haven't used yet uh 9.6 squared plus 18 squared is I don't know 20.1 squared once I find out that I don't know I'm in great shape 9.6 squared is 92.16 18 squared is 324. is going to be 404.01 for 20.1 squared if I add these two together I get 400 and [Music] 15.16 416.16 . yeah that would make sense uh and 416.16 is greater than 404.01 so since the sum of the two smaller sides was greater than the longest side squared I have an acute triangle which means all three angles are less than 90 degrees kind of tricky but you know when you break this all down you can kind of be like oh okay I see what that guy did there find the area of this triangle right here well the area of a triangle is one half of Base times height I have a base you know what I don't have I don't have a height so what I can do is I can realize that this is an isosceles triangle boom boom boom and if it's an isosceles triangle that means the height is going to go directly down the middle so why don't I cross you out and say that one of these triangles right here is going to be 4.8 nope 3.8 3.8 3.8 okay so how can I find that height I'll call you H well if I split this big triangle into two smaller triangles I now have a right triangle and if I'm trying to find the missing side of a right triangle I use an old friend called Pythagorean theorem a squared plus b squared equals c squared I'm going to call UC because U represent the hypotenuse of this right triangle right here I'll call you a and I'll call you B because it really doesn't matter which one's a or b as long as you call the hypotenuse C so I'm going to call a h so H for my height squared plus b which is 3.8 squared is going to equal C which is 5. squared 3.8 squared is 14.44 so I will write that out that's going to equal 5 squared which is 25. in order to get H all by itself I have to subtract 14.44 from both sides so minus 14.44 drop down the H squared and that's going to give us 10.56 now in order to get squared all by itself I have to square root it not squared all by itself in order to get rid of squared I have to square root both sides so that crosses out I'm dealing with triangles so I don't have to worry about this symbol right here I've talked about that in previous videos and if you're like what never mind uh I don't know the square root of 100 or of 10.56 is so I'm going to go off to the side and ask my phone and it's awful so three point it's two four nine six one five blah blah blah so I'll do 3.25 I think that should do it if it's 3.296 I think three point uh to five should do it 3.2496 that presents the height of my triangle so let's get rid of all this stuff now that I I'm rid of it what do I know well I now know my height is 3.25 ish I have my height I have my base let's plug it all in to the formula a equals one half of my base which is 7.6 times my height which is 3.25 I have zero clue what this is off the top of my head so I'm going to go off to the side and ask my phone again and the old phone says 12.35 so I'm not going to circle that most teachers would be fine with it but I'm going to say my area is through 12.35 I look up here for units I don't have any so I will say units squared because whenever you're given area you should give some type of unit even if you're not given a unit and when you're finding area it's always unit squared feet squared inches squared whatever okay but that's how we find the error use of Pythagorean theorem find the perimeter of the larger triangle all right let's see I have 42 26 uh oh this chunk is 11. but I don't have that part right there hmm well what I have is I have two right triangles inside a larger right triangle maybe even this might not even be right who knows it may not be it doesn't matter all I know is that this is a right triangle that's a right triangle so what I can do is I can find this missing side right here by using Pythagorean theorem once I find that missing side I could use Pythagorean theorem again to find that guy add all four of these sides up and I have myself my perimeter so let's do that let's find this missing side right here using Pythagorean theorem let's call this let's give it a name let's call it h because it's the height of this triangle that way when I use Pythagorean theorem twice I'm not super confused with the letters that I used so I am going to use Pythagorean theorem a squared plus b squared equals c squared now looking at this triangle right here this little one with h11 and 26 I know C has to be my 26 because C always has to be the longest side the side opposite of my right triangle it doesn't matter which one's a or b as long as the longest side is C so a is H squared so H squared plus b is 11. so 11 squared is going to equal C which is 26 so 26 squared H squared doesn't change 11 squared is a hundred twenty one that's going to equal 600 176. now in order to get H all by itself I have to get rid of squared and I have to get rid of minus or of regular 121 And I do that by minusing 121 I was going to ahead of myself got excited there so let's subtract 121 from both sides H squared is going to equal 500 and 5. hmm square root both sides and I have zero clue what the square root of 555 is so I'm going to go off to the side and do that so it's gross and I knew it would be gross that's kind of why I took a moment to figure out what this was because I thought I was going to get something pleasant and I didn't at all it's 23.558 so let's go with 23.56 so this guy is 23.56 all right that guy the height yippee now I'm going to do the same exact thing to find out this part right here so I'm just going to call it X all right I am now focusing on this triangle on the left where I have a height of 23.56 I have a hypotenuse of 42 and I'm missing a length of x so I'm going to use Pythagorean theorem again using different colors so I don't mix the two a B's and C's I will call you a again because you know why not I'll call you b as long as I call the longest side C so a is 32.56 23.56 squared plus b is I don't know x squared equals 42 squared before you reach for a calculator we know what 23.56 squared is it's 555. we just did that oh yeah plus x squared is going to equal 42 squared which is 1 764. subtract 555 from both sides subtract 555 from both sides and you're looking at 9 carry the one make that at two one two o nine gross is equal to x squared is X all by itself no squared's in the way so let's square root both sides no clue what the square root of 1209 is so I'm going to go off to the side gross it's 34.770677301 so I'll put 34.77 . this guy right here and I'll make it black is 34.77 so in order for me to find the perimeter of this whole triangle I have to add that side plus 26 plus 11 plus 34.77 42 plus 26 plus 11 plus 34.77 that's going to end up equaling 113.77 uh no units to worry about so units feet inches whatever just don't Square it because we're not finding area we're finding parameter pretty gross problem not a big fan took a while but uh yeah I got it I got it good find the missing side lengths so with this problem we are going to use special triangles those of you who are like I did geometry already and I'm using this as a refresher course don't do trigonometry here and if you've never taken trigonometry before and you're like the heck then we're going to use what's called special triangles there's two types of special triangles but the one we're going to focus on right now is the 30 60 90 triangle the 30 60 90 triangle is named 30 60 90 because it has measures of 30 degrees 60 degrees and 90 degrees when you're doing a 30 60 90 triangle the smallest side this side in between 60 degrees and the right angle is going to be a we'll just call it a in this case 2. the other leg is going to be whatever the smallest leg is Multiplied to the square root of three the longest side the hypotenuse is going to be whatever the shortest leg is Multiplied to two this is the rule for the 30 60 90 triangle the smallest is whatever it is to find the other leg you multiply the smallest leg by the square root of three and to find the hypotenuse you multiply the smallest leg by two so in order to find v v is going to be the other leg the smallest leg is two so V is going to be the smallest leg multiplied to the square root of three and that's it don't run your calculator don't type it out we're going to leave it like that U is much friendlier U is going to be the smallest leg times two so two times two let me get out my calculator is four so my smallest leg is two my other leg is 2 square root of 3 like that and my hypotenuse is two times my smallest leg two times two is four this is how you use the 30 60 90 triangle to find the missing sides find the missing side lengths all right what we have here is a 30 60 90 triangle there's a little rule that says if you're given a 30 60 90 triangle okay we'll call you 30 degrees we'll call you 60 and you're going to be my 90. you find out what the smallest leg is call it a the other leg is going to be a times the square root of 3. the hypotenuse is going to be 2 times a that's how you deal with the 30 60 90 triangle it's a shortcut okay those of you who are using this video as a refresher if you're thinking oh I'm going to use sohcahtoa don't you could but don't if you have no clue what sohcahtoa is pretend I didn't say that now what do we have we have the hypotenuse we're not starting out with the smallest side okay that is my smallest sign that is my a okay so I'm working my way backwards well let's use common sense okay if the longest side the hypotenuse is two times the smallest side then in order to find the smallest side we go in reverse divide 16 by 2 which according to my brain is eight and you're thinking oh this isn't bad I can find X easily uh hold your breath a little bit because this one's not going to be so pleasant actually I take it back I now have the smallest side I have the smallest side the smallest side is eight which means to find the other leg all I have to do is multiply the smaller side by root 3. so I'm done what I was initially thinking is I was going to take that 16 divided by root 3. and you might be thinking uh isn't it bad to divide by a square root it is because you have to do all these little extra rules called rationalizing the fraction but we didn't have to do that here because that was a little more clever again what I did is I was given the hypotenuse I know the hypotenuse is two times the smallest leg so the smallest leg is half the hypotenuse dunzo why is eight once I have the smallest leg I know that the other leg is just root three times the smallest leg so root three times eight is eight times or three and if I wanted to I could have found X by dividing by root 3 which would require to me to do a little bit of extra work but didn't so I'm done find the missing side lengths what I have here is a 45 degree angle and a 90 degree angle and Common Sense tells me you're also going to be 45 degrees this is a special triangle called the 45 45 90 triangle 45 45 90 gets its name from the fact that I have a triangle with a 45 degree angle another 45 degree angle and a 90 degree angle the legs here since this is an isosceles triangle or exactly the same so I'll call you a and I'll call you a as well the hypotenuse if you're thinking oh I'll just do Pythagorean theorem don't worry about it because after you do Pythagorean theorem you're just going to multiply the square root of 2 to the smallest leg that's kind of the trick that's why this is a special case right triangle so over here I have a leg of 2 times the square root of two well if one of my legs is 2 times the square root of 2 that means B the other leg is also going to be 2 times the square root of 2. now to find the hypotenuse which is a I'm going to take my smallest leg both of these are the same so they're both the smallest leg and I'm going to multiply it to the square root of 2. so the square root of 2 times the square root of 2 means you can multiply inside the square roots and get the square root of four oh I know what that is the square root of 4 is 2. which makes a equal to 4. so this is a special case triangle it's a 45-45 90. and when you have a 45 45 90 whatever one leg is the other leg is also and to find the hypotenuse you just multiply a leg by the square root of two and in our case we had to do a little bit of clever math but it really wasn't that bad and we're done find the missing side lengths all right I have a triangle that's 45 degrees that's 90 degrees which means you're 45 degrees as well now this is a special case triangle called the 45 45 90 triangle it's called 45 45 90 because the angles are 90 degrees 45 degrees and again 45 degrees which means it's isosceles which is good because whatever one leg is the other leg is exactly the same if I'm given a leg what I can do to find the hypotenuse is take that leg and multiply it to the square root of 2. so if you're thinking this is pretty simple just take a leg and multiply root 2 to it to get the hypotenuse I'm done great except this one's not that simple see I'm given the hypotenuse I need to find both legs X and Y are going to be the same thing so once I find X I have y the problem is is I'm starting out with a hypotenuse so if I'm given a leg and I multiply square root of 2 to get the hypotenuse then in order to get the leg from the hypotenuse I divide by the square root of 2 because I'm working in reverse order now when you have a square root and you divide by a square root we can shove 2 off to the side and divide both of these guys within the single square root so you get 6 divided by 2 inside the square root last time I checked 6 divided by 2 is 2 times the square root of three and so my X and my y are both going to be 2 times the square root of 3. so there are times where you use special case triangles like this one and you're given the leg and it's easy when you're given the hypotenuse you have to work in Reverse which means you divide by the square root of 2. and in our case it still ended up being a fine number if you're like oh let me whip out my calculator and find out what that is don't okay whenever you're dealing with with right triangles you're going to see square roots quite often don't lose sleep over it I certainly won't neither should you find X all right there you are let me see I have two triangles uh uh attached to each other X is way up here where the only information I'm given is 90 degrees and 45 oh that makes you 45. great but that's not enough information to give me any lengths here so let me go to this other triangle which is up 45 and 90. so you're 45. okay now there's a little rule when you're given triangles that are 45 45 90 that we can use and that uses the special case triangle 45 45 90. the 45 45 90 triangle says that if you have a triangle where your angles are 90 degrees and 245s if I give you a leg we'll call it a the other leg has to be the same exact thing because this is an isosceles triangle which means these angles are the same and the legs are the same in order to find the hypotenuse you take the leg and you multiply root 2 to it so whatever this leg is times root 2. so I have one length here and it's seven so since this is seven just for funsies I will say that that is seven does that help me no what will help me is the fact that this will be the hypotenuse of this bottom triangle here that's the hypotenuse 45 45 90 the hypotenuse is square root of two times a leg so you are 7 root 2. done nope because my next step is now that I have that little piece of information I could use this triangle right here and find my missing hypotenuse so if you're seven root 2 just for funsies I can say that you're 7 root two does that help me no but what I do know is that this is a 45 45 90. and according to the 45 45 90 rule if whatever one of the legs are I just multiply it to root two so X my hypotenuse is going to be one of the legs to the leg times root 2. so I have 7 times the square root of 2 times the square root of 2. if you're multiplying two square roots together just multiply the insides and keep that square root there so 2 times 2 is 4. I know what the square root of 4 is it's 2. and I know what 7 times 2 is it's 14. so X is 14. any units nope so that's my guy so I use special case triangles to find the missing side when I'm given two triangles glued together find X all right what do I have here x is way over here it's attached to this little triangle looking thing that has an angle of 45 and 90 oh which means U would also be 45 do I have any information about this triangle no I just know that it's a 45 45 90. great let's look at this other triangle I have the hypotenuse of this triangle it's 90 it's 45 and it's 45 again so I have 245 45 90s I know that 45 45 90s are special types of triangles okay it's a special type of right triangle that says if I that's supposed to be 45 45 90 poorly drawn but it says if I have a triangle with 45 and 45 and 90 degrees if I have a leg here the other leg is exactly the same because this is isosceles also the hypotenuse is going to be square root of 2 times a leg so if this was 5 5 root 2 . got it now how does that help me here well I have to work my way from 7 Root 6 all the way to X which means I probably have to find that line connecting it so what I'm thinking is I'm going to start out looking at this big triangle right here that is 45 45 90. I have the hypotenuse the hypotenuse is 9 root 6. I need a leg over here I said if I give you a leg then the way I find the hypotenuse is I multiply by root 2 well this is not simple because I'm not giving you the leg I'm giving you the hypotenuse which means I have to go in reverse so I'll call you H because it looks like it could be a height so I'm just going to call it h in order to find H which is a leg I have to take this hypotenuse and divide by root 2 because I'm working in reverse so my hypotenuse is 9 Root 6 I'm going to divide by root 2. now when you do the square root divided by a square root I can take this 9 here and shove it off to the side and put both of these numbers inside the same square root what that gives me is 9 times the square root of 6 divided by 2 3. so this value right here that H is 9 root three for funsies this is also 9 root 3 but it doesn't really help us now what I have is I have this triangle where I have to find X so I'm going to color it blue okay and what I'm going to do is just because I'm mucking this up I'm going to move it down I'm going to recreate the fact that this is 9 root 3 and I have to find X and you're a right angle you're 45 degrees you're 45 degrees I'm given a hypotenuse of 9 root 3. this is a 45-45 90. I know that if I was given a high Pi a hypotenuse and I have to find the leg I divide by root 2. again so X is going to be 9 root 3 divided by root 2. now here is the problem with this guy if I use the same process that I did here nothing's going to simplify I knew that I could do 6 divided by 2 and so I did this you're not allowed to have the square root of 2 in the denominator of a fraction so when you have the square root of 2 in the denominator of a fraction what you do is a process called rationalizing the fraction you multiply the top and the Bottom by the square root of 2. what does that do for us well the top becomes nine times the square root of three times two is six the square root of 6. anything there change nope the square root of 4 and if you're thinking why am I why is that help us I thought that was supposed to get rid of the square roots what's the square root of 4 two so nine times the square root of 6 over 2 is now a legal answer and so that's my X and if you're thinking oh can I simplify these no because one's in a square root one's not so this problem was much more complicated than a previous video that I had but the same idea is there it's just with this one I'm giving you the hypotenuse instead of giving you a leg so complicates things a little bit still tons of fun I enjoyed it find X there you is what do I have I have a big triangle this triangle glued to that triangle uh all right I have 9 Root 6 and X is way over here so X isn't attached to 9 Root 6 really in any way whatsoever what do I know X is attached to a triangle that has a 90 degree angle and a 60 degree angle which means you're a 30 degree angle oh probably going to end up using the 30 60 90 triangle so let's draw that out 30 60 90. the 30 60 90 triangle says that if I have a triangle with a 90 degree a 30 degree and a 60 degree whatever the smallest leg is I'll call you a the hypotenuse is going to be exactly two times that smallest leg the other leg is going to be the square root of 3 times the smallest leg so what I'm going to do is I'm probably going to end up using those facts for this guy but I can't even begin this guy until I figure out what I have over here my 9 times the square root of 6 is attached to this leg of this right triangle that has an angle of 60 degrees so that means you're 30 degrees as well so this is the ah the longer side of a 30 60 90 triangle so what I have to do is work backwards a little bit I have to work backwards to find this guy right here I'll call you a a in my case here represents the smaller leg of this right triangle now if I were to find the bigger leg of a right triangle that's 30 60 90 I multiply it by root 3. so if I wanted to work in reverse and say let's now go from the big leg to the small leg that means I have to take 9 Root 6 and divide by root 3. so again I called a this leg right here you are now 9 Root 6 divided by root three six over three is doable so what I'll do is I'll just move the nine off to the side there's a rule that says if you're dividing square roots just make it one big square root and divide on the inside so 6 over 3 inside the square root six over three is two so that length right there is nine root two 9 root two this is not going to be pleasant at all because now I have part of this guy right here now I'm looking at my smaller right triangle right here gross can't draw at all so what I can do is let me let me redraw it make it a little bit bigger make a little bit more pleasant to look at your 90 your 60 your 30. you are now 9 root 2. I need to find you I don't know you so just like this guy I'm given the larger leg and in order to find the hypotenuse of a 30 60 90 triangle I have to deal with the lar or the smaller leg I have the larger leg if I'm given the larger leg and I have to find the smaller leg what do I do let's call it B just for funsies B is going to be 9 root 2 divided by root 3 just like we did up here there's a problem though I could do six over three and make it go away I can't do two over three in Roots so what I have to do is I have to rationalize the denominator rationalize the fraction I have to get rid of that square root of 3 because you're not allowed to have square roots of 3 in the denominator of a fraction so the way you get rid of it is you multiply the top and the Bottom by the square root of three this is going to make you happy you'll see why in a moment okay B is going to equal 9 times the square root of 2 times the square root of 3 which is the square root of 6 when you're multiplying square roots you multiply within the roots and make it one bigger root the square root of 3 times the square root of 3 is the square root of 9. square root of 9 is 3. so B is 9 Root 6 over 3. 9 and 3 don't have square roots so I can simplify 3 divided or 9 divided by 3 and make it just 3 Root 6 because 9 over 3 is 3. so this guy my smaller side my smaller leg is 3 root 6. so if I have a 30 60 90 triangle and my job is to find the hypotenuse and I have this smaller leg I'm finding X now I take this smaller Leg 3 Root 6 and multiply it by two two is outside the square root so is three so three times two is six x equals six root six no units nothing like that we use special right triangles in order to find missing parts of two right triangles glued together find the value of the trigonometric ratio as a fraction so we're given this right triangle all three sides are given to us I have angles two but no information about that except the fact that you're 90 and it's asking us to find sine of C that's not sin C that's in Ohio Cincy Cincinnati anyway so uh if we are ever asked to find sine C or cos C or tangency you are going to use this little acronym so car TOA now what we do is we take a look at the angle that we care about and we look at the sides compared to the angle for example 35 no matter what's the hypotenuse 28 is going to be opposite of c and 21 is next to it which means adjacent so sohcahtoa tells us look if you're using sine you care about the opposite and the hypotenuse if you're using cosine you care about the adjacent in the hypotenuse and if you're using tangent then you care about opposite over adjacent now we're using so and so means the sign of whatever angle you're using in this case C is going to be the opposite side over its hypotenuse okay so the opposite side according to C is 28. 35 no matter what is your hypotenuse so sine C is going to equal the hypotenuse on the bottom and the opposite on top 28 over 35. and that's it that's just the trigonometric ratio I can't do anything with that can't simplify it we're going to leave it like that it says leave it as a fraction so we're going to leave it as a fraction so sine of C in this case represents 28 over 35. find the value of the trigonometric ratio as a fraction so we're given a right triangle all three sides are there no information about the angles except that's 90 and we're being asked to find cosine of C not cos cosine of C whenever you're asked to find sine of something or cosine of something or tangent of something you use the little acronym so Ka TOA what that does is it tells you look if you're finding sine or if you're finding cosine or if you're finding tangent you care about either opposite sides a hypotenuse or an adjacent side now if we take a look at C okay I'm being asked to find the cosine of C I'm using this one because this is saying the cosine worries about the adjacent side and the hypotenuse so I care about finding cosine C now if I'm asked to worry about the adjacent the adjacent side according to c means the side next to it that's not the hypotenuse 27 is right next to it the hypotenuse is 45. don't be like oh well 45 is adjacent too yeah it is but 45 specifically is the hypotenuse so this stands for cosine of your angle equals the adjacent side over the hypotenuse adjacent over the hypotenuse in other words cosine of C is the adjacent side which is 27 over 45. now in this case it looks like I can divide both of these by nine so I can simplify this further and say that cos C is three-fifths now depending on the teacher that you have they might not care about that they might just be happy that you notice that it's 27 over 45 but if you notice that you can simplify it simplify it but there we go there we go there we go uh like And subscribe find the value of the trigonometric ratio as a fraction tens that's what I get in the summer what we have here is we have a right triangle all three sides are given 90 is given nothing else though no other angles but we are being asked to find the tangent of angle Z now if you're asked to find a trigonometric ratio you are going to use this clever little phrase so katoa now what this does is this tells you listen if you're trying to find sine you're going to use this one if you're trying to find cosine you're going to use this one if you're trying to find tangent you're going to use this one we're trying to find tangents so we're going to use this one now there's multiple things that we need to worry about here I care about Z this one Z okay that's my Z Angle now what TOA stands for is the tangent of whatever angle you are trying to find has to equal o and a what is O and A O stands for opposite and you always put it over the other letter which is adjacent so this would be adjacent over hypotenuse opposite over hypotenuse but in this case we're trying to find the tangent which is opposite over adjacent so if you're like what's opposite over adjacent well we compare the opposite side and the adjacent side to Z so the opposite side is going to be the side opposite of Z which is 21. the adjacent side is going to be the side next to it and if you're thinking 35 and 28 are both next to it yeah but 35 is a hypotenuse we want this side next to it that's not a hypotenuse so not U and yes you so the way we set this up now is we say all right tan Z is going to equal the opposite side which was 21 over the adjacent side which was 28 and you might notice 21 over 28 does that simplify it sure does 28 21 divide them both by seven three quarters and that's my answer tan Z is three quarters I've heard of gen Z but tanzi that's all we're trying to find here find the value of the trigonometric ratio as a decimal we're given three right triangles and we are being asked to find three ratios as a decimal whenever you're asked to find the trigonometric ratio you are going to use this cute little phrase so Ka TOA now sohcahtoa means if you're trying to find sine then you have to worry about the opposite side and the hypotenuse you're trying to find cosine then you worry about the adjacent side and the hypotenuse if you're trying to find tangent then you worry about the opposite side and the adjacent side let's do one at a time all right sine of Z there's my Z when you do sine of Z you care about sine Z equals the opposite side over the hypotenuse now well let's let's do one thing at a time hypotenuse now we compare Z with these numbers we're going to say all right well if you're just looking at this and not really worried about the letters and you're like which one's opposite which one's adjacent look at Z the opposite to Z is 30 so my opposite is 30. my hypotenuse is always 50 because 50 is going to be opposite of the right angle 30 over 50 simplifies 2.6 so sine Z is going to be 0.6 in this one problem right here uh you know what Jesus died for that our sins uh cosine Z all right so there's a new Z right there and cosine is going to use K which I believe is slang for friend now cos Z has me caring about the adjacent side adjacent means next to over the hypotenuse so if you look at Z over here and if you're thinking isn't 12 and 15 adjacent is in 12 and 15 both next to Z they sure are but 15 is the hypotenuse which would make 12 the adjacent side now 12 over 15 this fraction is not near as friendly as this but if I notice that both of these are divisible by 3 that becomes four over five and I know four over five is the decimal 0.8 so cos Z is going to be 0.8 let's get a new color and do a new problem 10x now Tan x where's my X there you is Tan x is going to use TOA so tangent X is going to be the opposite side over the adjacent side okay now looking at x the opposite side of X is 30. I can handle that I have two adjacent sides 40 and 50. oh but 50 is opposite of the 90 degree angle which makes 50 my hypotenuse so my adjacent side is going to be 40. 30 over 40 is the same as saying three quarters and I know three quarters is 75 cents so that's what 10x is lots of puns in this one I like it I enjoy it hopefully that gets me more than 10 views but uh yeah there they are trig values written as decimals fun fun find each angle measure to the nearest degree what they're asking us to do is find each of these three letters here's the problem for each of these three letters there is a trig function in the way sine is in the way cosine is in the way tangent is in the way so I have to get rid of each of these trig functions legally now the way I do that is I use the inverse trig function so what I'm going to do is I'm just going to rewrite this guy give myself some space I'm going to have to do that with all three of these okay and I'm definitely going to need a calculator for this one you guys can't do these on your own back in the day people have used slide rules those were a nightmare now my issue currently all I did by the way was rewrite that my issue currently is sine is in the way you can't divide sine from both sides what you do is you take What's called the inverse sign some people call this the arc sine same thing inverse sine what the inverse sine does to a regular sign is it gets rid of it okay so now b is all by itself the problem is I have to go to a calculator and find out what the inverse sine of 0.4848 is so if you're using like a TI 84 plus you're looking for that inverse sine button if you're using Desmos you have to hit functions and use that button so I'm actually going to go off to this side here I use my Desmos and I'm now typing in I'm not typing in anything I'm hitting the inverse sine of 0.4848 make sure your calculator is set to degrees there's something called radians it looks like r a d you don't want that even though it sounds cool because it's rad you don't want that now I typed it in my calculator and I actually think I'm set to radians I am so I made a degrees and according to this I have 28.999 and it wants me to round it to the nearest degree so 28.999 is 29 degrees okay so what that means is if I were to run to a calculator and type out what sine of 29 degrees I would get 0.4848 blah blah blah blah blah it's never going to be a nice number most of the time unless you're dealing with like zeros and and 90 degrees and stuff like that but most of the time it's going to be very unpleasant I'm going to write this guy again cosine a Leave myself some space 0.43 7431 in order to get rid of cosine I have to inverse cosine some professors some teachers call that arc sine inverse cosine both sides that crosses the coses out and I run to a calculator I type out inverse cosine of points 0.7431 and I get 42.0038 so I'm going to go with 42 degrees okay uh same thing here all right I'm going to write out tan W equals give myself some space 19.0811 in order to get rid of tan I'm going to inverse tan both sides so in verse 10 in verse 10. that gets rid of tan some people call that arctan and now W is going to equal whatever the inverse tangent of what was it 19.0 811 I get 86.999999 86.9999999 is awfully close to 87 degrees now I am on an iPad using a Desmos app make sure whenever you're in geometry class pre-calc is a different story but when you're in geometry class make sure your settings uh like you might have to hit that little wrench thing make sure your settings are set to degrees okay it says it right there if you use radians you're going to get a very small number every time for this maybe not too small for that but you're going to get a very small number if you're using like a TI-84 or like a Casio or some kind of smart uh calculator graphing calculator it might say something like rad next to degrees like if you hit the mode button it'll probably say rad in degrees you want the degrees okay so make sure your calculator is set but that's how you do it if you are told sine of an angle equals stuff inverse sine it inverse coset inverse tanet and uh you'll get that dunzo use sohcatoa to find the missing angle I'm going to call this angle X just for Math's sake and I'm supposed to find that missing angle well I have a right triangle that's extremely important I don't have the other angle but I do have two sides it's telling me to use sohcatoa so this is what I'm going to do I'm going to take a look at X 42 compared to X is my hypotenuse 40 compared to X is the opposite side so I am going to choose the three of these that involve the hypotenuse and the opposite side and it's U so what I do whenever I'm using so is I get a needle and thread I say sine of my angle which I called X is going to equal the opposite side over the hypotenuse okay so I don't know X so sine X equals the opposite side which is 40 over the hypotenuse which is 42. excuse me math teacher goat you left a whole lot of space there is there a reason yes I have to find x x here represents the angle measure sine is in the way so just like Christ did for us I have to wash this sin I'm sorry sign away by inverse signing both sides in order to get rid of the sign you inverse sine both sides and that cancels out the sign on the left the problem is is I have to find out what inverse sine of 40 over 42 is which I have zero clue so you always not always but most of the time in Geometry you're going to have to run to a calculator which I'm on my iPad I'm on Desmos right now and I'm typing out inverse sine of 40 over 42 and I get 72 points uh two four seven two so I'll write all those out two four seven two now this is an angle measure X is nowhere in the original problem so I'm not going to circle this for two reasons again because X is nowhere in the original problem and I don't have my degrees so I'm going to write out 72 point and if you want to round this to two decimals find two five degrees but that's my missing angle measure okay and I use sohcahtoa to do it I had to do inverse trigger trigger them trigonometry trigonometry to do it and I landed on that guy again if you well when you have to use a calculator make sure your calculator has a mode set to degrees which means if you're using like a TI 84 plus or something like that you hit mode and you make sure it's in degrees if you're using Desmos you click the little uh wrench looking thing and make sure it's set to degrees otherwise it's going to give you a completely wrong number but that's the right number and I'm happy use sohcahtoa to find the missing angle first things first you're being asked to find the missing angle and I know if we have to use sohcahtoa we're going to need a calculator so make sure your calculator is set to degrees so if you're using Desmos you're going to hit the little wrench and set it to degrees if you're using a graphing calculator make sure you go to mode and set it to deg and not r a d all right I'm being asked to find this question mark right there and I'm going to call it X I'm given a right triangle and I'm given 20 and 13. now I have to compare 20 and 13 to x 20 no matter what is my hypotenuse 13 is right next to X which means 13 is going to be my adjacent side no opposites here so you know just like if I were calling my friend I'm going to use code this car is slang for friend uh is cosine of your angle equals the adjacent side over your hypotenuse so according to what I have I'm going to write out cosine of the angle that I need to find which I did call X is going to equal the adjacent side which is 13 over the hypotenuse which is 20. if you're thinking wow you sure unnecessarily left a whole lot of space here I did leave a whole lot of space but not unnecessarily my job is to find X that missing angle cosine is in the way the way you get rid of a trig function is you inverse trig both sides so I'm going to inverse cosine that guy and I'm going to inverse cosine that guy which is definitely going to require that calculator I was talking about the inverse cosine and the cosine cancel each other out I run to a calculator and I type out the inverse cosine cosine to the negative 1 of 13 over 20 which my I'm using Desmos my Desmos says 49.45 so 8 3 blah blah blah blah blah foreign if you get a really small number now by the way if I look at this you're never supposed to look at a picture and say yeah that looks right but it does look right that does look like it's in the 40s as far as angles go if you never set this to degrees and it was set to radians it would give you a very very very small number and you would know that you're wrong set it to degrees now I'm not done yet for two reasons first off I never used X in the original problem I created X so I don't need X second this is an angle so I have to put degrees and I am going to round this to two decimal places so my final answer is going to be 49.46 degrees that's my guy all right so in this case X 20 and 13 were my hypotenuse and my adjacent which is a and H use cosine use inverse cosine type it in your calculator or Desmos or whatever it is that you're using and you're good to go use sohcahtoa to find the missing angle so I have to find that angle right there I'm going to call you X now whenever you're doing right triangle trigonometry and that's what this is because sohcahtoa stands for sine cosine and tangent which is trigonometric functions you have to always use a calculator not always but like 99 of the time make sure if you're using Desmos you hit the little wrench and have it set two degrees if you're using a graphing calculator a scientific calculator make sure it's set to degrees and not radians radians are something completely different you'll find out in precalc now I'm looking at this missing angle here that I called X I have no other angle so it's not like I can subtract from 90 or 180 or anything like that that's no good so I have to use these two sides right here what I have to do is compare 27 and 38 to X now the obvious one is 27 is opposite of that 38 is next to it so is that my adjacent or is that my hypotenuse well this is my hypotenuse so you're my opposite for sure and you're my adjacent for sure and I don't have information about the hypotenuse unless I wanted to do Pythagorean theorem but that's a waste of time so opposite adjacent TOA now we're going to set up what TOA means TOA means the tangent of an angle is going to equal the opposite side over the adjacent side okay so here I have tangent of x equals the opposite side which I called 27 over the adjacent side which I called 38. wow I sure left a lot of space there didn't I I did and I did that on purpose because I have to get X all by itself and the only way you get X all by itself is to get rid of that tangent so we get rid of tangents by taking the inverse tangent of both sides so we write out the inverse tangent of both sides that allows us to cross out those tangents over there now X is all by itself I am running to my graphing calculator right now and if you're using Desmos you hit functions if you're not if you're using a graphing calculator you probably have to hit second tangent and then the little tangent inverse pops up and then you type out 27 over 38 and according to the Desmos that I'm using I get 35.934795 blah blah blah blah blah blah now I'm not done I'm not going to circle that for two reasons reason number one I never used X in the original problem I created X just because also I don't have a decimal and that's just that's too many numbers so I'm going to round this off to two decimal places and say that my answer is 35.39 degrees all right and you can look at this and say yeah that looks about right if for some reason you did this and you got a ridiculously small number that's because you have it set to radians and that's not rad it's 90s slang for really cool find X using socatella all right I have a right triangle I have X which represents what looks to be a hypotenuse so I'll label you hypotenuse I have an angle right here and I have a side but that's all the information I have so it's not like I'm given another side I can do Pythagorean theorem I have the angle I have two sides and that's it well sohcahtoa helps us when we only have two sides or only need two sides and we're given information about an angle so I'm looking at my angle 72 degrees and I'm comparing what x is to 72 degrees well I already said it's my hypotenuse so what is six six is directly next to my angle which is my adjacent sign now before you're like well doesn't isn't X also adjacent it is but that's my hypotenuse and stop talking like that so I need to choose the one of these that uses the adjacent side and the hypotenuse and why don't I call my friend or my Ka uh Paul Walker one time said that in Too Fast Too Furious like 20 years ago he's like forget about it could if you've seen it you know all right so if we're using Ka we are going to set up cosine of my angle which I'll call a because I'm afraid to call it X is going to equal my adjacent side over my hypotenuse and now I plug in what I know I plug in cosine of my angle which I know my angle we did previous videos where we didn't know my angle but now I do is going to equal my adjacent side which I know is 6 over my hypotenuse which is X so in a previous problem I was trying to find the angle and we needed to use inverse cosine not here what I do is I run to whatever calculator I have or Desmos or whatever I have and make sure your calculator is set to degrees so if you're using Desmos hit the little wrench thing and go to degrees if you're using like a TI 84 make sure you go to mode and make sure it's in degrees and you're literally going to type out cosine of 72. which I am actually doing that this very second it's giving me the fraction [Music] 0.30901 so I'll just call it 0.309 that's going to equal 6 over X my job is to solve for x so this is what I'm going to do X is on the denominator I don't like that so I'm going to create a denominator for you and look at this out of proportion you don't have to do it this way this is just the way I'm doing it and if you don't like it watch like friggin Khan Academy or something they're just not cool and funny whenever you have a proportion crisscross applesauce so I'm going to multiply 0.309 to X which is 0.309 X and you're going to mult set that equal to 6 times 1 which according to my calculations is just six you divide both sides by 0.309 divide both sides by 0.309 so I'm actually going to type that out right now 6 divided by 0.309 and I get 19 0.41747 so depending on how picky I mean I didn't tell you where to round it to so let's round it to two decimal places 14.42 no units to worry about don't have to worry about inches or feet or anything like that so that's how you find X where we're finding a hypotenuse using trigonometry awesome find X using sohcahtoa so socatoa means like trigonometry stuff sine cosine tangent I'm given this right triangle here which is the only times I can use sine cosine and tangent in geometry and I have to find a missing side not find an angle find a missing side I have an angle and I'm going to compare these sides to that angle right there X compared to 37 is an opposite side 12 compared to 37 is adjacent and the side that I don't care about is a hypotenuse so I need to pick the guy that cares about opposite and adjacent and that's TOA so what TOA stands for is tangent of an angle equals the opposite over the adjacent side make sure when you do this your calculator or Desmos is set to degree so go to mode or go to that little wrench thing and make sure you're set to degrees but I'm going to set this up according to this tangent of my angle I have my angle it's 37 is going to equal my opposite side which I don't know over my adjacent side which is 12. do you know what tangent with 37 is I don't I doubt you do so I'm typing it in a calculator right now tangents 37 according to Desmos is [Music] 0.753554 an unnecessary amount of decimals but I always like to write more than I need So Tan 37 turns into that nothing happens on the right side until now is X all by itself no it's not 12 is being divided from it but this is just an algebra one problem multiply both sides by 12. so I'm going to take that ugly number that my calculator has I'm going to multiply 12 to it and I get Point nope 9.04 uh two six blah blah blah is going to equal x so let's flip this around clean it up a little bit X is going to equal let's round to two decimal places so it's not five or above give it a shove 9.04 leave it as that that is my X looking at the picture that looks like it would make sense I don't have to worry about feet or inches or units of measure here there you have it there you have it just make sure when you do these you have your calculators or make sure Desmos or whatever it is that you're using is set to degrees otherwise you're going to get terrible answers find X using socatella all right so X is over here I'm I'm given a 30 60 90 triangle and Satan is tempting me to use that but I'm not going to maybe I will I don't know maybe I'll just give in to my Temptations but I can't even do anything right now I'm just given a hypotenuse and 60 degrees that don't help me but over here on this triangle I'm given 39 degrees I'm given 38 that's it oh if only I could find out what that side is if only I can find out then I would have a side here 60 degrees an X and that would make life a lot easier so that's what I'm going to do I'm going to look at this triangle right here I have an angle of 39 degrees 38 compared to 39 degrees is my adjacent side I have to find this side this side which I'll I don't want to use x so I'll call it y this side is opposite of my 39 degrees so I'm going to use the trig function that involves the adjacent side and the opposite side and that's TOA when we set up TOA we say tangent of my angle is the opposite side over the adjacent side tangent opposite adjacent so I'm going to fill in what I know tangent of 39 is going to equal the opposite side which I don't know over the adjacent side which is 38. my issue right now is I have zero clue what 39 degrees is make sure if you use Desmos if you use a graphing calculator you have your mode set to degrees otherwise this is not going to work so I'm going to go to a calculator right now and I'm going to type out tangent of 39 degrees and it's telling me that it's 0.80978 so 809 we'll just make it 9 8 right there okay I'm rounding it a little bit and that's going to equal y over 38 okay 1039 turned into that is why all by itself no but it's over 38 and I know from algebra If I multiply both sides by 38 y will be all by itself so let's take that number that I typed into my calculator let's multiply it by 38 and it's giving me 30.7717 so I will say 30.77 is y now Y is really not called y I just called it y so I'm going to get rid of U and I'm now I'm also going to get rid of my words because I don't want to confuse anything okay what I do know is that this is 30.77 I also know that this is a 30 60 90 degree triangle but I'm not going to use that I'm just going to use sohcahtoa again okay I'm going to look at 60 and compare this guy an X to 60. that guy is opposite compared to 60. X is my hypotenuse no matter what don't be like it's also adjacent yes but it's my hypotenuse so I need the guy that uses my opposite and my hypotenuse and that would be sine so the way you set up sign is you say sine of your angle is going to equal the opposite side over the hypotenuse and now we fill in what we know sine of 60 is going to equal the opposite side which is 30.77 over uh the hypotenuse which is X so let's go to a calculator and type out sine of 60 I do know what it is because I teach pre-calculus but we're going to type it out anyway it's 0.866025 so I'll leave it like that that equals 30.77 over X now X is in the denominator so I'm going to use this little trick where I'm going to turn this into a proportion so I can do crisscross applesauce do I have to do it this way no I can multiply both sides by X but I'm not gonna I have it set up as a proportion now and so I'm going to multiply diagonals so 0.866 X well can't forget that is going to equal 1 times 30.77 I think I can handle the math for that one in order to get X all by itself we divide everything by 0.866 divide everything by 0.866 and let me just do this off the top of my head just kidding I'm typing it into my calculator 30.77 divided by 0.866 and I get 35.5311 so we will say x is 35.53 because it's 1 1 and I'm not going to worry about those ones no units of measure no inches or anything like that to worry about I am solving for x so I'm allowed to say x equals but yeah kind of a crazy problem uh you know we had to use sohcahtoa twice but listen man I I love it so if I had to use it 50 times I'd use sohcatella 50 times making these videos I basically have so if you appreciate that uh smash that like button find the area of the entire triangle there is an area formula okay that involves trigonometry it involves signs but I'm not going to use that I'm going to use this area formula right here half of Base times height that's a b half of Base times height I have my oh I and you know what else I don't have my base either so this problem is just very unpleasant now I'm going to cheat a little bit all right and please don't please don't get mad at me I'm going to cheat a little bit this guy right here okay this guy is a 30 60 90 triangle I know with 30 60 90 triangles whatever my smallest leg is my hypotenuse is going to be two times that and my second leg my longer leg is going to be a times root 3 of that so if I'm given the smallest leg multiply it by root 3 and I get the longer leg I have the longer leg so if I want to find this height right here I'm going to divide by root 3. so I'm going to go to my calculator because I'm going to need my calculator a whole lot anyway I'm going to divide 29 divided by the square root of 3 and I get 16.7431 so I'll put 16.74 16.74 all right uh uh I'm not going to clear that I'm going to leave that there now that's great now I have my height but I still don't have the entire base this is what I do have I have this triangle on the left I have an angle I need to find this I'll call you X and I have a side next to it so I'll call you my adjacent side I'll call you my opposite side and I'm going to use sohcahtoa now if I have my adjacent side and I have my opposite side that uses the TOA in sohcahtoa and so you set up tangent of your angle which is 52 is going to be the opposite side which is I don't know over the adjacent side which is 16.74 I run to my calculator and I type out tangent 52 make sure your calculator is set to degrees and not radians and I get 1.27999 so I'll put 1.28 and that's going to equal x over 16.74 do I have my x yet new but if I multiply both sides by 16.74 I will have exactly what I need 16.74 good job giving myself a lot of space Nick that's my real name my real name is not math teacher goat I just call myself that because I'm trying to get more followers but what I'm doing is I'm running to my calculator right now I'm typing on 16.74 times 1.2 weight and I get 21.42622 so I'll do 21.426 X represented that missing side right there so let's get rid of it and now call it 21. 0.426 so let's write this triangle out let's let's draw the exact same triangle as you can see it's perfect I know the height it's 16.74 I know if I add these two numbers together I get uh my entire base so 21 plus 29 is 40 so this whole thing is going to be 40.426 so to find the area I use this formula and do a equals half of my base which is now 40.426 times my height which is 16.74 I'm not even going to pretend that I know what this is so I'm going to my calculator right now and I'm typing it out while I'm doing that make sure you smash that like button make sure you crush the Subscribe button am I sounding like a YouTuber now um times uh 16 point what was it 7 4 16.74 and I typed out the wrong number 300 38.365 so I'll call it three seven I don't have a unit so I will call it square units that's the uh area of this whole thing now there's a way that you could do this a whole lot faster using a calculator and sign and stuff like that but that's what I did here because that's what I was asked to do if you want to know how I do it faster watch the next video on your playlist and don't forget to uh smash that like button give me some of that sweet YouTube money use socatella all right km is 10 root 3 km all right this whole thing is 10 root 3 the measure of LMN which is this guy is 60 degrees fine Ln okay so what I have what I have is a rhombus I have to find at the end of the day l n okay so I'm going to use the fact that this whole thing is 60 degrees this whole thing is 10 root 3 the whole thing but I only want this part of the triangle which is half of that so half of 10 root 3 is 5 root 3 okay so let me get rid of that 60 because I now don't want that whole 60. I know that this length right here is 5 root 3. you would be 30 because if this is a rhombus then the diagonals bisect the Angles and I'm going to use this triangle right here to help me find ZN once I find ZN multiply it by 2 that gives me Ln so I'm going to use that triangle which was 30 degrees [Music] this guy was 5 root 3 and I need to find this guy once I find this guy that gets me ZN multiply it by two and I get the whole thing so what do I have I have 30 degrees now oh the devil is saying use special triangles boy use it it says you socatoa so get thee behind me Satan I'm going to use what the rules say I use because the rules are not meant to be broken so I have 30. 5 root 3 is opposite of that no it's not it's adjacent are you down with OPP oh [Music] all of that all of that just to be stupid all right 3 root 5 is adjacent X is opposite so I need to use the one of these three that deals with adjacents and opposites and you're my guy so tangent of an angle is going to equal my opposite side over my adjacent side let's plug in what I know tangent of 30 is going to equal my opposite side which is X over my adjacent side which is 5 root 3. so tangent of 30. run to your calculator make sure it's set to degrees and type out 10 30. you can hit functions or you can just hit the tan button in a regular calculator tangent of 30 is 0.57735 so I will write that out 0.57735 and that equals x over 5 root 3. is X all by itself new I have to multiply both sides by 5 root 3. so this is me doing that that allows me to cancel that out X is going to equal now let me run to my calculator again and take this guy and multiply it so hit the times button five root three wouldn't you know it I get a perfect five so if God is smiling down on us now this is five right here now again if we didn't have that and if you know special right triangles you would know that if I give you the smaller leg the longer leg is going to be 5 root three so this makes sense now ZN is five Ln is 2 times 5. so Ln is 10. I made a rhyme no units to worry about no inches no feet nothing like that fun problem use so katoa so sine cosine or tangent all right oh my gosh TQ is 15. there you are which means you're also 15. I don't know if that's going to matter but I'm just doing a TR this whole thing is 20. I'm not going to write that out yet XR is 11 find the measure of angle x t s x t s all right all right all right all right all right uh all right so this whole thing is 20. this whole thing is 20. which means you're nine okay so this problem is actually not near as bad as I thought it would be now where did I get 9 from if the whole thing is 20 and you're 11 then you're 9. oh 9 11 never forget so what I'm going to focus on is this triangle right here that triangle right there where I'm being asked to find this angle that's a right angle you're nine and you're 15 and I'll call this T because it's there's a t there okay now if my job is to find angle T and I'm using sohcahtoa I have an angle and I need to see what 15 and 9 are compared to that so 9 compared to T is right next to it and it's not my hypotenuse so it's my adjacent 15 is my hypotenuse so I need the guy that cares about the adjacent and the hypotenuse and it's co as Paul Walker once said In Too Fast Too Furious what's KO Ka is cosine of your angle is going to equal the adjacent side over the hypotenuse now I fill in what I know cos t equals my adjacent side which is 9 over my hypotenuse which is 15. wow I left a whole lot of space there stupid me stupid you because T is not all by itself we have to get rid of cosine and in order for us to get rid of cosine we take the inverse cosine of both sides so I needed that space so I could do that the cosines on the left are gone okay I have to run to a calculator or Desmos or whatever and find out what inverse cosine of 9 over 15 is while I do that make sure you understand that your computer your your whatever it is that you're using calculator I'm trying to multitask and I'm doing a terrible job make sure your calculator is set to degrees otherwise it's going to give you a terrible answer this is probably going to be a terrible answer anyway but uh yeah it's not it's it's not pretty 53.1301 so we'll stop at 53.13 and that gives me that angle right there X t s so I'll finish this answer by saying the measure of angle s t s equals 53.13 degrees okay use area equals half of little a little B sine Big C to find area now I'm given that triangle right here and you're probably thinking what's this what's little a little B sine c what does that mean if I were to take just any regular Old Triangle and you're going to see this a lot when you get to pre-calculus your angles are going to be capitalized it doesn't matter which one's a which one's B which one's C in this case okay it doesn't matter at all the sides opposite of the angles are going to be your lower case side so a is over here lowercase a c is right here opposite of that is lowercase C and B is right here opposite that is lowercase b so if we're going to use little a little B and Big C then we care about side angle side so it's basically you're using in a sense SAS which I use SAS all the time in my videos but in this case we're using side angle side to figure this out oh look side angle side so I can call you little a Big C and little B so let me get rid of this and let's find the area the area and I'm not going to use a for area because I'm gonna I don't want to make it look like the a that I just have written out so area is going to equal one half of a which I called eight B which I called six doesn't matter which one you're a little a and little b r because you're just multiplying anyway so sine 87. so I'm literally just going to type this straight into my calculator this whole thing make sure you have your calculator or Desmos set to degrees otherwise it's not going to give you a pleasant answer so 6 times 8 times I had that backwards doesn't matter sign of 87 and it's telling me 23.967108 blah blah blah blah blah blah blah foreign so let's finish this problem out by saying that the area is 23 Point Let's do 9 6 7 centimeters [Music] squared okay so that's using the trig area formula uh to find the area of a non-right triangle if you're given an angle in two sides as long as the sides are attached done fun number one name each figure we have five shapes and not only are these shapes they're solids they're 3D shapes three dimensions so I have to name each of them there's several types but I think we're going to look at three main types uh here if they end up in a point at the very top imagine that's a 3D shape it's a pyramid of some type if they go straight up and create some type of shape as you can see poorly drawn there that is a prism and then there's the ones that are round like that guy and that guy will will end up doing uh those in a second now if it's a pyramid when or a prism you are going to name it by saying the name of the base and then pyramid or the name of the base and then a prism so if it's an octagon turning into a pyramid it's an octagonal pyramid if it's a pentagon turning into a prism it's a pentagonal prism so we're going to name all five of these based off of that information I'll leave that there I think I'm going I won't need a ton of space what we have here is a prism because all of these go up and create a base on the bottom and a base on a top the base on the bottom and on top is a pentagon so this is a pentagonal prism which I think I accidentally said a second ago but you know what who needs to repeat itself pentagonal not to be mixed up with Pentatonix oh acapella group all right we have a pyramid like you see here that is a triangle on the bottom so that is a triangular pyramid and this is neither pyramid nor prism it's a ball what's that called that's a sphere if you don't know what a sphere is then why are you in this course why are you taking it why are you watching these videos now this is probably the weirdest looking one all right what you see is you do see something that is prismic in nature but it's off to its side it's actually tilted up sideways see it looks like this guy over here the base is on the bottom and the base is on the top in this case the base is on the side and the side and the bases are actually trapezoids so this is a trapezoidal prism you just have to pretend it's on side the hardest one of all of them is this last guy right here and it's called a cone well that's it we named each figure and again you're looking at naming the type of Base shape it is and then either pyramid or prism unless you have a sphere unless you have a cone and I guess anything that would be like these two would be a semi-sphere which is a half um but all of them otherwise would be one of these so yeah these are these are your solids find the volume what we have here is we have a cylinder now I'm actually going to use the volume of a cylinder find the volume what we have here is a cylinder now this is a geometry course and I'm assuming that if you're in this course you have already taken 8th grade math or maybe you're in eighth grade right now but the volume of a cylinder is usually taught somewhere in eighth grader before it and it's given to you by pi r squared H now where this formula comes from pi r squared is the area of the circle all right since this is a cylinder the bases are circles the H is the height so it's basically you're saying let's just pretend like this circle is like a pizza and H represents how many pizzas you're stacking up and you're just multiplying all of them together so all of this is connected now it's just a matter of doing some unpleasant math now Pi is the number Pi 3.1459 blah blah blah blah blah R is the radius eight is also a radius if it was going all the way across the circle it would be the diameter but since it's from the center to the outside eight is the radius oh happy day seven is also just the height oh happy day for this one so it's just a matter of typing out into your calculator pi times eight squared times seven so I did that I went off to the side and I got 1407.4335 so I'll leave it at point four three now what you don't want to forget is your unit this is volume and it gives you kilometers so you are going to use cubic kilometers area is measured in square units regular old distance regular old length and width are measured in kilometers feet whatever but this is volume and this is a 3D shape so you are using three-dimensional units kilometers cubed but that's it right there that's the volume of a cylinder nothing crazy something I'm sure you've seen before find the volume of this 3D shape right there all right so this is strange what we have is a prism because whatever your base is on the bottom is also the same on top so let's go with that first the volume of any prism any prism whether it's a cube or something crazy like this a hexagonal uh prism is going to be the area of the base which we denote by capital B times the height okay the area of the base times the height uh now the base is a regular hexagon okay so if we were to break this down further and create a new formula the volume is going to be one half times the apothem times the perimeter and then times H over here so the volume of a hexagonal prism or a hexagon I'm sorry yeah the volume of a hexagonal prism is one half times the apothem times the pyramid times the H now if you're like what the heck's hypothem isn't that an animal that only comes out at night no that's a possum an apothem is the length that connects the center of whatever regular polygon you have to the middle of the side of the polygon it's basically like a perpendicular bisector that connects to the center so the apothem in this case is 3.5 feet so if I wanted to find the volume of this it's one half times the apothem is 3.5 times the perimeter now what we have is we have a hexagon which means there's six sides all of these sides are four feet so the perimeter is going to be four feet six times so that's four not point four times six now we're going to multiply that to the height which appears to be three feet now I'm not even going to pretend to try to do this in my head although it's possible I'm just going to go off to the side and type it all in my calculator and Mr calculator says 126. now my unit of measure is feet so since we're dealing with volume we're going to make sure we say cubic feet remember when it's volume its cubic units and in this case we get a pretty nice number because the numbers that they gave us weren't so bad but the volume of any prism is the area of the base times the height and we were able to break it down a little bit further since the bases were rectangular rectangular polygons I'm sorry regular polygons so many so many things that I have to say just like my words get jumbled up but there you go 126. find the volume of this shape right here now what we have and what's going to be difficult to view is we have a triangular prism now usually we view a prism as something that stands up like this well what if I connected these and made it look like that now I have myself a triangular prism and so this guy is that guy so if I were to turn this into that if I were to take this and kind of tilt this over sideways I would see that this has a height of three that would be my height it has a side length right here appears to be three you appear to be four and you appear to be five okay now why does all this matter well the volume is going to be the area of the base times the height okay now this is a triangle okay so the area of this triangle I'm going to view from here okay if this is a triangle I'm going to replace area of base with what the area of a triangle is one half times base times height now remember the height that I'm looking for is the height of the triangle the height that I want for here is the height of my we'll say P for prism okay so let's just make sure that when we do this we're mainly focusing on just the triangle so one half of Base times height of a triangle so I'm looking at this triangle this triangle if I were to take this and stand this up like so this triangle would have a base of three a height of four and this hypotenuse of five which one of these don't I need the five so if I were looking at the triangle it's one half base which is three times height which is four so that takes care of my area of the triangle part now the height of my prism goes back to the picture that I drew over here which is also three so I should be able to do this in my head um one half of three times four is going to be one half of 12 which is six and six times three is eighteen that's supposed to be an eight now the unit that I have here is feet so I'm going to make sure that I say feet cubed because this is volume and volume is measured in cubic units but there you have it the volume of a triangular prism is 18 cubic feet in this case done find the volume of this shape right here what I have is a trapezoidal prism but the trapezoidal prism doesn't quite look like a prism because usually when you think of a prism you think that the base it has to be like on top and on the bottom as well and here it's not well if you were just to imagine that this is laying flat on its side right and I know my my picture is not going to look great at all I'm just going to try my best so there's the top oh [Music] you get the idea trapezoid on the top trapezoid on the bottom prism part if I were to take this and lean this flat and fall this down uh I would have this guy be five that's a five right there you would be from top to bottom you would be 4.3 so that is the height of the trapezoid uh the height of the actual prism is for uh the small part on Top is three and the large part on the bottom is eight so I have a I have a prism with that information the bases are trapezoids and the trapezoidal information is a height of 4.3 a top of three and a bottom of eight okay it also gives me like this this slant looking thing of five I don't think that will matter and then the four is the height of the trapezoid so let's let's put a formula together the formula and this is a prism and the formula for any prism is the area of the base times the height of the prism itself okay now what I want to do is I want to remind myself of what the base is it's a trapezoid and the the area of a trapezoid is one half times the base of the trapezoid in parentheses base sub 1 plus base sub 2 close it height of the trapezoid now I'm going to put height of the trapezoid right there so you can see it's the height of the trapezoid some teachers might um put this as one half height and then the base it doesn't really matter in fact that's what I should have done but I accidentally misspoked and started drawing out the b and I just kind of went with it now that's going to be multiplied to the height of the prism okay so I will put that there unfortunately I have to get rid of my terrible terrible terrible drawing so I just have to remember that when I'm talking about the red here I'm talking about just the trapezoid and when I'm going to ask myself for the height of the prism it's four because again if I were to lay this flat it would be four meters tall so let's put this all together V equals one half now it doesn't matter which one's which or your base of the trapezoid so I'll call Base sub 1 3 and base sub 2 8 again doesn't really matter the height of the trapezoid is 4.3 and the height of the prism is four so I can't do this in my head so I'm going to go off to the side and figure out what this is the old Apple watch says it's 94.6 now this is volume and the units are meters so I'm going to make sure that I say meters cubed because it's volume but that's a trapezoidal prism and you find the volume of a trapezoidal prism by using this formula or you could just jump straight to that formula whatever floats your boat whatever tickles your pickle but that's what I got for this one 94.6 baby find the volume what we have here is a cone now if you're taking this geometry course chances are you're in high school maybe you're in eighth grade maybe you're in seventh if you're that good but at some point you've probably been asked to find the volume of a cone whether you have this formula memorized or not I do so let me take care of you the volume of a cone is one-third pi r squared h okay one now if you're looking at this by the way and you're like wow that's awfully the same as a uh cylinder it is because a cone is a cylinder with a bunch of sides hacked off that's why it's one-third pi r squared H it literally would take you three of these to fill up a cylinder of the same exact dimensions that's for a different day now what I have here is 1 3 pi r squared H what I have is I have my height my height is 18. what I don't have is my radius this 18 it has to do with my diameter so I don't care about 18 I care about half of it which is 9. so what I do is I take those numbers I plug it into my formula V equals one-third pi 9 squared H which is 18. I have no clue what that is pi is a nightmare anyway so I'm going to go off to the side and I'm going to ask Siri what it is and she says 1526 point oh I lost it eight one four zero so I'll do eight one four all right volume miles so I'm going to make sure I call it miles cubed okay because volume is always measured in cubic units so again this is something very standard for a seventh or eighth grade class depending on what state you live in in eighth grade you do cones or in Pennsylvania you do cones in eighth grade and you know that's where I live the Commonwealth so there you have it Breezy identify each solid given its net now a net is a basically an unwrapped version of a solid figure for example if I were to take a cylinder like so pretend that's a can of soup a nice delicious can of soup and if I were to unwrap This and like kind of cut this part out I would have myself a nice little circle if I were to cut down here and wrap it out or unwrap it I would have myself a rectangle and keep in mind there's an invisible Circle here that you really can't see if I were to pull that down I would have myself another Circle so that is a net of a cylinder okay this guy I know what you're thinking you're like that's a star no it ain't that's a pentagonal prism or a pentagonal uh pyramid what was the first one prism would be wrong pentagonal pyramid because the one base is a pentagon and if you were to take all of these triangles and fold them up they would end up being a point thus making your pyramid this guy if you were to take this half circle looking thing and wrap it around that Circle here you would get yourself a cone okay and if I were to take this guy Fold It Up take this guy fold it up and wrap these around I would have a rectangular prism so back in the day when I taught eighth grade I was forced to teach this to the students and they it was it's hard to visualize 3D objects on a 2d uh space which is what I'm doing here and I'm sure you can find a thousand YouTube videos that show you what this looks like what Nets look like using wonderful animation I'm not about that I just use like you know computers and and simple stuff I'm a simple man with simple needs copy the measurements onto the net Annette is a basically if I were to take this picture this guy right here and take a bunch of scissors and just like cut out the Pentagon on the top and cut out the Pentagon on the bottom and kind of like cut these and fold these around that would be the net of it okay it's a flattened version of a 3D shape my job is to turn that into that well what I have what I see first is probably the grossest thing this is the apothem now the apothem is essentially the radius of a regular polygon so I'm going to put a dot right in the center right there and have it go to one of the sides right there draw my 90 degree angle and that is my 8.9 it appears to me that each side of the Pentagon is 13. and if you really wanted to go crazy so are you and so are you and you're 13 and you're 13 and you're 13 and you're 13 and 13 and 13 and 13 13 13 13 13 13 13 13 13. but I'm not gonna do that that is the height we'll call it five which means your five your five so is you so is you so is you so is you okay but that's me copying the measurements onto the net Bang copy the measurements onto the net all right what I have here is I have a square pyramid and I'm taking this apart and basically if I were to take a scissors and just cut that up at the very top and all the triangles were to kind of fall down to the side that's the picture I would get so let's draw the measurements that 20 and that 20 represents the side of this Square so I will just say that you're 20 and I'll mark all of these the exact same because why the heck not that 26 belongs to the height of a triangle so I'm going to put a 90 degree thing right there I'm going to call you 26. now that 26 has a name and it's called slant height whenever you have a cone or a pyramid and you're probably being asked to find a volume the slant height is like the height of the diagonal side of whatever shape you're dealing with now what that looks like flat is the height of a triangle in this case but that's all I had to do here copy the measurements onto that dunzo funzo oh these are not fun find the lateral area and surface area first off what's the difference between the two the lateral area is basically the area of the sides not including the bases so first off you have to make sure that your shape is sitting exactly the way it has to and lateral area would be like well let's find the area of a box if I were to take off the top and the bottom and we only care about the sides so in this case the lateral area is going to be that rectangle that rectangle and that rectangle in the back the surface area is all of the sides so in this case the rectangle the rectangle the rectangle the triangle on the top and the same exact triangle on the bottom now if you wanted to do it that way or if you wanted to split everything apart and be like all right well then the lateral area is let's find the area of U plus the area of U plus the area of U absolutely nothing wrong with that nothing wrong at all but if if we want to save a little bit of time there are formulas for prisms now that's supposed to be a straight line and I failed miserably let's do the lateral area first okay and the formula for the lateral area for any prism is going to be and I'll use l for lateral the perimeter of the base times the height okay the perimeter of the base times the height of the shape so in in my case uh the lateral area is going to be eight times or I'm sorry plus six which is 14 plus 10 which is 24. the height of my shape is four and so the lateral area for this guy is 96. I can handle the math on this one now this is kilometers and we're talking about area so the lateral area is kilometers squared because with area you square your units that is the lateral area now what I want to do is I want to find the total surface area of my shape here okay so the total surface area which I'll call SA and I know what you're thinking hey that's what I call my friends I call them essay no not for this surface area is going to equal 2 times the area of the base we've seen that capital B when we've done um what should we call it's volume before plus the perimeter times the height all right so first things first we have a triangle for a base okay so if I wanted to find the area of the base I'm looking at my triangle one half times base times height if I were looking at this triangle right here it would sit perfectly fine on top of that eight so one half of Base would be one half of eight times height would be six so half of 8 is 4 4 times 6 is 24. okay so each base has an area of 24. so my Surface area is going to be 2 times 24 Plus Perimeter times height well wait a minute I did that already I got 96 well let's not over complicate things 2 times 24 is 48 and 48 plus 96 is 144. a nice number again we're dealing with area so our units of measure are going to be square kilometers okay again with these guys you can split it up in different parts and add up each side pretend you have decisions and cut out all five sides here and just add them up and add three of them and add five of them but whatever I did it I did it using the formulas that's easier for me I did it I did it it's done find the lateral area and surface area of this guy a cylinder now if you're wondering well what's the difference between the two the lateral area is the area of the sides now imagine okay if imagine if this was like a can of soup right and I would cut the top off I would have myself a circle if I were to take the sides and kind of like cut it up here right and unravel it I would get myself a rectangle and if I were to take the bottom and cut the bottom down I would have another Circle so the lateral area is this rectangle right here the surface area is the top the sides and the bottom that's the difference lateral area is once you sit your shape down the exact way it's supposed to sit it on top of its base the lateral area is the area of the sides and the surface area is the area of everything sides top and bottom this is a cylinder so I'm going to split this up into two sides so I can find the lateral area and we'll do the lateral area over here and then we'll do the surface area over here the lateral area is found by doing and it's very similar if you watch my video for uh what did I do a triangular prism it's very similar you could say perimeter times height so the same exact thing but I know that this is a circumference perimeter means circumference which is 2 pi r height so instead of saying perimeter height I'm going to do 2 pi r height problem number one I'm not given uh my radius I'm giving my diameter so I'm going to cross you out and make you a seven okay go now let's keep the same color bilateral area is going to be 2 times pi times radius uh times the height which was eight don't know what that is so I'm going to go off to the side and ask Siri so she said 351.8583 so I'll do 351.86 kilometers this is area so kilometers squared that is my lateral area so if I were to unravel this huge cylinder that is seven kilometers in radius and eight kilometers high a very realistic shape it would be pretty big pretty big now I've got surface area so I'll put sa for surface area and the surface area now includes the top of the bottom so without looking up a formula sheet if it's going to include the top and the bottom then the surface area S A is going to equal 2 pi RH which is the side 2 pi r h Plus two areas of circles so 2 pi r squareds so that shouldn't be too crazy to look at that shouldn't be too wacky to understand I know part of this already I know that 2 pi RH 2 pi RH is 351.86 2 times pi r squared is going to be 2 times pi 7 squared so let me go off to the side Siri says it's 307.8 uh uh 7X so let me add these two together asking Siri to do it for me and I get 659.736 kilometers [Music] squared mm-hmm so yeah I mean you know let's just say that you were making like some kind of Tower and you you needed it to be eight kilometers high and uh 14 kilometers long and you're like oh man I wonder how much uh how much material I need you're gonna need roughly uh 659 uh kilometers of uh squared of material which is roughly the size of Rhode Island probably not Rhode Island but you know like maybe like a city like maybe like Philly so a lot find the lateral area and surface area of this pentagonal prism first things first a lateral area versus the surface area if I were to take this shape and sit it on its base which it currently is the lateral area is just the side so these rectangles right here and all the rectangles are the same so the lateral area shouldn't be too crazy to do what is going to be crazy to do is the surface area we'll get to that the surface area is let's add up all five sides and the top and the bottom that's the surface area so let's find both of them let's find out what the lateral area is first and you find the lateral area of any prism by doing the perimeter of the base times the height of the prism now the perimeter of the base is found by saying well look we have a pentagon there's five sides each side is 12 yards long so 5 times 12 60. the height of this prism is eight so 5 times 12 is 60 and 60 times 8 is 480 uh and this is yards squared and again this makes sense because if I were just to take one of these rectangles I would get 12 because it's 12 yards long times eight I would get uh what's 12 times 8 72 76 76 and multiply that by five the numbers seem to work out just fine what's not as fun is the surface area my essay I don't have a lot of essays anymore I have four kids if you're trying to understand that joke essay is another word for friend now for any any prism your surface area is going to be uh your lateral area which is nice because I could just kind of replace that plus 2 times the area of the base so so far not so bad my lateral area my perimeter of the base times the height was 480. I can handle that I can handle it just fine plus two times now this is a regular polygon all right a regular polygon's area is found by doing one half so times one-half that makes life easy times the apothem and if you're like apothem is isn't that a creature that comes out at night and is ugly and has creepy eyes no that's a possum the apothem is 8.3 it's basically like a radius of a polygon a regular polygon so one half times the apotham which is one half times eight point three times the perimeter of my polygon which I already know to be 12 times 5 which is 60. so my Surface area is going to be 40 or 480 plus 2 times a half well I don't need that anymore 8.3 times 60 I'm going to go off to the side and figure out what that is Siri says it's 498. add these two together and you get 978 yards squared because area is square yards um there you have it pentagonal prism lateral and surface area fun fun fun fun fun fun fun fun fun fun find the lateral area and surface area of this pyramid right here well the lateral area of a pyramid is the sides the surface area is the sides plus your base your one base so if you're thinking oh this is kind of similar to when they were prisms kind of but not exactly the same okay not exactly the same so let's start out by doing the lateral area first okay the lateral area of any prism is found by doing the and I'll put l for lateral area one half of the perimeter of the base times what is called the slant height now that's a pretty looking l because that pre-looking L represents the slant height the slant height is the height of one of the sides if I were to cut this up and kind of stand one of these triangles tall and if you're thinking oh it's basically the height of a triangle the slant height is the height of the triangle okay so it's very very very very very very very similar to one half base times height which is the area of a triangle it's just the undoing instead of one half base times height I'm doing one half perimeter of this whole thing times height of one triangle okay so lots of similarities here so let's plug in what we have one half is one-half the perimeter is I have one two three so this is a pentagonal prism each side is eight feet so the perimeter is going to be eight times five which is 40. the slant height is 10.6 so to find the lateral area I'm going to do one-half times forty times 10.6 which is 212. it's 20 times yup uh feet feet squared because of area being square units now Common Sense will tell you that if I'm finding just the sides to give me the lateral area and the only difference is now I have to add my one base then why don't I write out the formula I'm not going to go to a formula sheet to find the area of the whole thing the surface area okay it's going to be my lateral area which is one half P times slant height plus the area of my one base all right so I know that my lateral area is 212. the base is a regular polygon so you find the base by doing one half times the apothem times a perimeter and if you like apothem isn't that an animal that comes out at night and has creepy eyes and sometimes gets hit by my car no that's a possum an apothem is a line connecting the center of a regular polygon to the side of the polygon as long as it creates a 90 degree angle like a radius if you will so my Surface area is going to be 212 plus one half times my apothem which is 5.5 times my perimeter which I already found out earlier to be 40. okay a little bit more math 212. plus one half times 5.5 times 48 which is 132 and you add these guys together and that's a three and that's a four and that's a four and this is feet squared yeah pyramids Egyptians find the lateral area and surface area of this guy right here which is a cone I'll tell you one thing right now I I don't like u16 because you're the diameter I like the radius I care about radii in this class friends and I care about eight this guy refers to a slant height this the the height of the side here and if you might be thinking oh man that means I have to use like what Pythagorean theorem to find the height of the cone maybe now lateral area what is a lateral area that's the area of just the side of the cone not the circle on the bottom just the side of the cone the formula for the lateral area is pi times R times l so nothing to do with the height of my cone thank the Lord Pi is Pi R is eight L is 23.4 I'm going to go off to the side and plug that in my calculator and I end up getting 588.1061 so I'll do 588.1 now the difference between lateral area oh by the way oh no units so I'll put units squared units squared it just looks better even though my handwriting on this problem is kind of sloppy now the difference between lateral area and surface area is the lateral area only cares about the sides the surface area includes the base so don't be like oh let me look it up don't the surface area is your lateral area which I already know so pi r slant height plus the area of one Circle and the area of a circle is pi r squared so fortunately I didn't have to worry about the height of this cone at all I thought I was going to but didn't I already know that Pi RL is 5 8 8.1 plus pi 8 squared which according to my phone is nine point that can't be right what's pi times 8 squared yeah that's much better 201 619 so I'll do 0.1 and then you add these guys to get feather and that's what you get that's what you get when you add them together Paramore what's my unit you oh unit squared see how I made that into a perfect U flawless gross lateral area surface area not that much fun find the surface area of a sphere well the formula is not too bad the formula for the surface area of the sphere is 4 pi r squared where R represents the radius which I have so it's just a calculator exercise surface area is going to be 4 times pi times seven squared and 4 times pi times seven squared is 615.75 21 centimeters squared gross Denzel find the volume of this guy a sphere uh if you ever took eighth grade math or seventh grade math you've probably had to do this before so this isn't super new and here's the formula for it maybe you remembered it maybe you didn't but here's problem number one this is an R this is my diameter so if I take half of it to get my R half of 15.6 is 7.8 so that's my r so I it's just a calculator exercise four thirds times pi times 7.8 cubed let me say that again so Siri hears me four thirds times pi times 7.8 cubed is one thousand 987.7987 I'm going with all the digits that Siri gave me uh it's inches so inches cubed y cubed because this is volume and you do units cubed for volume all right volume of a sphere nothing new nothing crazy are the two figures similar if so State the scale factor figures are similar if you take all of the corresponding sides make a fraction out of all of those sides and the fractions are the same for example 12 matches up with 40 so 12 over 40. that has to equal 24 over 80 24 matches up with 80. so 24 over 80 and looking at this probably so far so good and 18 has to match up with 60 so so divide by 4 divide by four so that's three over ten divide by 4 I could divide by eight and divide by eight and that's also three over ten and divide by 6 divide by 6 and that's three over ten so the nice thing about doing it this way is we get a big fat yes for this one and not only do we get a yes this gives me my scale factor from small to big so we can write it out as three to ten I believe more often than not they want you to do small to big when they ask you for scale factor uh I know since or they want you to go from left to right so left to right here would be three to ten either way it will probably be specific and if it's not always go left to right or top to bottom or just say what it is but yes three to ten that's that guy uh uh since I have a whole lot of congruent stuff going on here I don't have to worry so much 18 matches up with 54 so I'll do 18 over 54. that's got to match up with six over 21. okay 18 and 54 I could divide them both by nine and that I can divide them both by 18 and that gets me one third 6 over 21 is I can divide You by three I could divide You by three two over seven ah no these aren't similar these aren't similar at all shame but that's how you do it very similar to if you have flat shapes if you have 2D shapes you just have to do a few more things if you want to do 3D shapes find the missing value two solids no picture I've got the surface area of solid one and this surface area of solid two I have the volume of solid one no volume for solid two this guy bigger than that guy big small now these are similar figures so I'm thinking ratio thoughts but I can't just this can't be just easy all right I'm going to compare the volume to the surface area and the way that I'm going to do that is I'm going to create ratios of big to small figures in other words the volume on the big solid is one three three one two and the volume on the small solid is I don't know call it X the surface area on the big guy is 1088 and the surface area on the small guy is 425. now we can't just end here we are now saying that volume is equal to surface area no it's not if you want to break down volume and surface area to its simplest most regular length I would have to cube root volume and square root surface area in order to get these guys equal to each other so similarly I have to cube root that guy and square root that guy in order to get things equal to each other all right so the left side doesn't change I have a big cube root one three three one two over X too much line okay uh 1088 divided by 425 is 2.56 now that's good because when I write it up here the cube root again I'm not changing anything here 13 3 1 2 over X that's going to equal the square root of 2.56 I believe it is 1.6 I use Siri I secretly use Siri now this cube root is just in the way how do I get rid of a cube root Cube everything Mark Cuban that cancels things out cancel cancel now I have one uh 13 312 over X is going to equal 1.6 cubed which is 4.096 why did you write that all the way up there so I can put it over one and make it a ratio and now I can do crisscross applesauce so I am going to do crisscross applesauce 4.096 x equals one third thirteen thousand three hundred and twelve divide both sides by 4.096 and thirteen thousand three hundred and twelve divided by 4.096 is three thousand two hundred and fifty now my goal was to find the volume kilometers so kilometers cubed these problems are not pleasant in any way whatsoever at all ever on the planet ever at all but we did it I feel good name the Ark made by the given angle all right so I'm giving these pictures and there's ways that you can name arcs okay an arc in a circle is basically if you were to look at the outside of the circle if I were to go from A to B I could name that arc Arc a b but what if I add all these different letters and what if I want to be more specific like for example if I were to say Arc AC which way am I going do I care about the right side A B C or ADC well I can name these guys like if I want to go if I want this portion of the circle right and I want to name that Arc I would call that Arc and here's my arc symbol Arc a d c if I wanted just this one which is called a minor Arc because it only involves two dots I would call that Arc b c or rxcb so you know by calling this Arc BC I know that I'm going right from B to C and I'm not taking this long way around so that's how you can name the arcs so when I'm looking at this guy right here the name of the arc made by the angle fqe here's F here's Q here's e well I have three dots on my circle but I'm going straight from F to e so there's two names for this one I can either call it f e or e f I'm not including D because I'm only going in this short route to get from FTE I'm not including the whole circle over here similarly angle one is between I and H and I don't care about J so I can either call this hi or I can call this eh either way I'm not going to include the J so when I I'm reading these I'm looking at the letters and that's kind of telling me telling me the direction that I'm going and which part of the circle that I care about okay so that's how I name arcs using angles name the central angle of the given Arc a central angle is the angle that basically its vertex is the center of a circle this area is called a sector not that it matters these are arcs okay now I'm given Arc ml so it's basically saying I want to go straight from M to L should I go this way no because otherwise I would have snuck a K in there this is just ml so I care about right here and the angle created by me going from M to L is just angle one okay nothing to think nothing to overthink pretty simple okay likewise again not being really creative with my letters here uh ml just goes from here to here again okay now that is this angle right here I'm not including K so m q l would be the name of this angle I could call it lqm it really doesn't matter either one mql or lqm uh both of those work but the central angle is basically your vertex is the center pretty simple find each measure first things first I'm going to find the measure of Arc cfd before we talk about that I have a lot going on with these circles so let's try to understand things if I have a central angle right here a central angle means it's vertex lives on the center of the circle and if I were to create an arc and we'll call you a we'll call you B we'll call you C if I were to say well let's call angle ACB 80 degrees whatever the central angle is the arc measurement is exactly the same thing so if an arc measure creates a central angle AB I have six of those the central angle is going to have the same exact measurement a c b or BAC or BCA I mean Basketball Club association uh so 80 is 80 in this case now when I go over here cfd is just more complicated than I need cfd means I'm going around here and around here and also here and also here and if you're wondering why didn't they call it CBD or cbfed or c e d or d f well basically this is just saying I need to go from C to D but I have to include F that's why I'm not just including this little chunk here now what I have is basically all of this right here problem number one I don't know you and I don't know you but what I do know is FC appears to be a straight line and if FC is a straight line then these two angles are going to be supplementary which means they add up to 180 degrees and if these two angles are 180 degrees then you're 45 which also means vertex angles you're 45. oh so the measure going from C all the way around to D is going to be 45 plus 135 plus 45 Plus 81. so 45 plus 135 plus 45 Plus 81. is 306 so the measure of angle c f d uh is 356. uh or 306 I kind of 40 and slip that's also true that's very true but what that means and most importantly is that means the measure of Arc cfd is 306. so I caught myself I mean I was accidentally naming the wrong thing but by saying this I can kind of like pretend like I know what I'm doing and be like yeah I was trying to do this the whole time which I am let's see if this one's a little bit more friendly the measure of angle m i j m i j is this angle right here I don't have any angle measurements I have this Arc measurement and Arc JK is 55 which means it's central angle created by it what is that i j i k i is also 55. so if you're 55 and this is a straight line these guys add up to 180. so the measure of angle m i j is 180 minus 55 which gives me 25. 125. degrees so I'll put that up here to keep things consistent the measure of angle mij is 125 degrees which means the measure of angle of Arc MJ Michael Jordan is 125 degrees and so on and so forth so central angles being matched up with their arcs everything's the same that's one of the nicer Parts about circles solve for x all right what do I have I have this guy which is a problem is called a ratio a radius this guy is a diameter I have this angle which equals that angle if this is a diameter that means this angle is 180 degrees if these two are equal that means you're 90 degrees so the way I can set this up is X Plus 93 is 90 degrees it's a one step or just subtract 93 from both sides and X is negative 3. now don't be like negative 3 you can't have negative measurements X is not a measurement it's just a variable so that if I wanted to find X I would be like negative 3 plus 93 oh this is 90 that arc length is 90. yay I did it so that's how you would do a problem like this not too bad not too bad at all find the given measure I have to find the measure of angle VST which is U now what I know is the arc measurement of Vu is negative 27x minus 3 which means the central angle created by that Arc is going to be the same exact thing also since this angle is congruent to that angle both of these are negative 27x minus 3. and last but not least this central angle creates this Arc negative 37x plus 2 which means you're that guy right there so this right now just saying each of these individually trying to find things isn't going to help me find X and in other words help me find this angle VST right here but what I do know is I have all three angles that create an entire circle so why don't I add up all the Angles and set it equal to 360 because the circles angles add up to 360. so let's do this angle right here negative 27x minus three let's say it again so I could put a parenthesis in front of it I can distribute but I'll just write it out negative 27x minus 3. now this guy minus 37 X plus 2 has to equal 360 Degrees okay now let's combine like terms negative 27 minus 27 minus 37 is negative 91 so negative 91 X negative 3 minus 3 is negative 6 plus 2 is negative 4 so minus 4 equals 360 degrees now it's a two-step equation I'm going to add 4 to both sides I'm going to add 4 to both sides so negative 91 X is equal to positive 364. I'm going to divide both sides by negative 91. and X is going to equal negative 4. am I done no because my job is to find the measure of angle VST the measure of angle VST is this guy which happens to be the same as negative 27x minus 3. so if I wanted to find the measure of angle VST I'm going to do negative 27 replace x with negative 4 minus 3. negative 27 times negative 4 is positive 108 and 108 minus 3 is 105. so the measure of angle v s t is 105 degrees there you go not too bad not too good but I like this stuff this stuff's fun for me fun fun fun find the length of this segment indicated all right when you're looking at a circle you have a diameter a diameter is a line that goes completely through a circle you have a radius which is a line that connects the circle to the end of the circle and you have a chord that just basically goes from one point to another Point those are your three lines mainly that you're dealing with that just live solely within the circle what I have is I have a radius which is 3.2 I have this line that if I were to keep it going would also be the radius but more importantly my radius goes through a chord there's a little rule that says that if your radius goes through a chord it bisects that entire chord and when you bisect something X is going to equal three whatever this is is going to equal that guy so with all this math x equals three so nothing here to overthink nothing here to just do too much with x equals three dunzo find the length of this segment indicated the segment that I care about right here is X I'm giving all this information diameter another diameter here's a chord I'm also given the fact that here's a radius radius of radius and they're all 9.7 now that's important not the fact that I give you three radii I don't care how many radii I get by the way I did say radii not radio radii okay video kill the radii star that's a song look it up but regardless radius is 3.7 I care about X but what I do know is again and it says it three times the radius is 9.7 so that red line that I just drew is 9.7 so if I wanted to find X X Plus 5.1 equals that entire radius 9.7 so if I subtract 5.1 from both sides X is going to equal 4.6 no units to worry about no units at all but there I have it I mean you should be able to look at this and be like well if that's the reduce don't I just do 9.7 minus 5.1 yup and that's it that's all you have to do with the guy like this I just showed work because it takes more time and I need that YouTube money find the length of this segment indicated so the segment that I need right here x is what I need to find let's see what I have I have a diameter the whole diameter is 8.18.8 uh that would make this the radius which if the diameter is 18.8 oh this is good news that means the radius is going to be half the diameter so that whole red line is going to be half of 18.8 which is 9.4 why is that important because what I need to find X is a part of my radius and I'm given the other part so X plus 4.2 is my radius which is 9.4 so in order to find X I just subtract 4.2 subtract 4.2 and X is going to equal 5.2 and if you're wondering then what's that 7.8 for nothing there to scare you there to throw you off that's it we're done find the length of the segment indicated uh oh this looks like The Devil Wears Prada logo uh those of you metal heads I played a show with Devil Wears Prada when I was in a band back then that's right I was in a band this voice does not get its own forget it uh all right so this is what I need to do let's not overthink it X represents a radius what is this guy right here a radius so why don't we call you X okay because they're the same thing why am I moving that X over there well because I am given two pieces of information about this guy which is clearly a right triangle and if I'm trying to find what appears to be the longest side of a right triangle and I give you the two shorter sides why don't I just use Pythagorean theorem to do that which is what I'm going to do so here's Pythagorean theorem I can call UC and it doesn't matter which one's a or b as long as the longest side is C so I have 8.3 squared plus 4.4 squared is going to equal x squared now let's do a little bit of math 8.3 squared is 68.89 according to my calculator 4.4 squared 4.4 squared I'm using Siri my calculator thought I just squared which is a fun game now I need to add these two together 68.89 plus 19.36 it's going to be 88 .25 that equals x squared in order to get X all by itself I have to square root both sides and remember usually when I square root both sides you're thinking positive or negative thoughts not when you're dealing with length the square root of 88.25 is going to be 9.3941 according to Siri uh do I have any units to worry about nope so all I have to do is just say x equals that guy right there it doesn't tell me where to round it so I'm keeping all four of those numbers I'd like to try to see you stop me I'd like to see you try and stop me I give up like And subscribe find the length of this segment indicated I need to find that x value right there now I have a long ways to go but this is how I'm going to set it up I see a radius of 15.1 I see a 12.8 over here and this radius right there a radius takes a chord and cuts it in half so this radius cuts that cord in half which means whatever this guy over here is that guy is also 12.8 my job is to find X if this was 5 then I would just do 15.1 minus 5 equals x but I don't know what this is ah but you know what this is connected to a right triangle I have a leg I have a hypotenuse so why don't I call you B because if it's a leg of a right triangle then let's just call it B I need to find B once I find B 15 minus 1 minus it gives me my X so let's use Pythagorean theorem a squared plus b squared equals c squared the longest side has to be C so 15.1 fits that I'll call you a because I already called BB and so I have 12 12 point what is that what is that 12.8 hmm 12.8 squared plus b squared equals 15.1 squared 12.8 square is 163.84 15.1 squared 15.1 squared is 228 0.01 [Music] now in order to get B all by itself I'm going to subtract 163.84 from both sides 228.01 minus 163.84 is 64.17 now in order to get B all by itself I have to square root both sides and the square root of 64.17 is a little over eight it's actually 8.01 it's a shame it couldn't just be eight but it's 8.01 so B this guy right here is 8.01 now again the whole purpose is I know this is a radius I'm trying to find X so in order to find X I can do the radius 15.1 there it is minus the chunk that I just found uh 8.01 15.1 minus 8.01 is 7.09 so that's my X so no uh units to worry about just numbers guns out find the circumference and area of a circle something that you learned way way way back in the day but we're bringing it back circumference is the same thing as perimeter the formula for circumference is 2 pi r in this case r is 4.2 so the circumference and I'm going to say this so that my phone hears me 2 times pi times 4.2 6.3893 26.3893 Series has given me all these numbers so I'm going to use them circumference is distance so it is just yards so don't worry about yard squared yards cubed nothing like that at all area is area and the area formula is pi r squared the same R so the area is pi times 4.2 squared which according to jobs is 55 .4176 yards squared because it's area circumference area stuff you saw in Middle School bringing it back find the circumference of a circle with an area of 64 Pi miles squared all right well the area formula is pi r squared I have to find the circumference which means I have to find the radius of this circle now if the area is pi r squared and it's giving me the area then I'm going to say 64 Pi is Pi R Squared first things first if we divide both sides by pi I'm about to do the pi is on both sides cancel out oh happy day that leaves me with 64 equals the radius squared I need the radius how do I get rid of that squared square root both sides normally when I square root both sides to solve an equation I attach a positive or negative symbol but I don't care about negative so I only care about the radius equaling 8. so the radius is eight and now that I know the radius is 8 I can find the circumference the circumference is given to us by the Formula 2 pi r in other words 2 times pi times eight which is 50.2654 Miles fun solve for x what we're dealing with here is what's called an inscribed angle okay we're used to seeing central angles an inscribed angle is an angle where all three parts the vertex primarily is attached to the circle now don't lose sleep if I were to say a b C the inscribed angle okay so the measure of angle a b c is going to be half of the intercepted Arc half of AC so if AC was like I don't know 60 you would be 30. okay so the way I'm going to view this is this guy this inscribed angle is going to be half of 192. so I made that really big now I have to get rid of it I'm going to set up the equation 31x plus 3 1X plus 3 is half of 192. so 31x plus 3 hasn't changed yet half of 192 is 96. now I have a two-step equation minus 3 minus 3 cross you out 31 x equals 93 divide both sides by 31 divide both sides by 31 and x equals 3. okay which is what I was supposed to do find X don't need to go any further but if I wanted to check my work I could do 31 times 3 is 93 93 plus 3 is 96 which is half of 192 what I was trying to find in the first place fantastic find the measure of angle nlm n l m is this guy right here okay now let's do this let's do this uh I have an inscribed angle okay and it looks kind of like this remember an inscribed angle so whatever the angle is the angle is half of the arc that it creates so nlm is going to be half of 7x minus 10. here's the problem I don't know what 7x minus 10 is no clear no clear no clue here's what I do know this is a straight line which means this Arc right here Arc l m n is 180 degrees so 13x minus 10 plus 7x minus 10 equals one hundred and eighty so if I set that up then I will be able to find out what x is and I'll be able to find out what that angle is so 13x minus 10 plus 7x minus 10 equals 180. let's combine like terms 13x and 7x make 20x negative 10 and negative 10 make negative 20. and I equal 180. now it's a two stepper add 20 add 20 and 20x equals 200. divide both sides by 20 x equals 10. am I done no sorry because again what I know is the measure of angle l n l m is going to be half of 7x minus 10. so let's say that the measure of angle n l m is half of 7x minus 10. well wait a minute I just found out that X is 10 so 7 times 10 minus 10. which means the measure of angle nlm is going to be half of 70 minus 10 which is 60 and half of 60 is 30 degrees that's how you do it okay inscribed angle stuff and inscribed angles are half of the angle or intercepted Arc that it creates and I had to do a little bit extra in order to get to that answer but I had a great time doing it and I hope you did too find the measure of Arc fgh which is this whole thing right here well let's draw a quick picture let's make sense of it usually and this is what I see by the way inscribed angle inscribed angle the two pieces of information that I'm given are all inscribed angles so let's quick talk about inscribed angles super quick I'll call you a I'll call you B I'll call you C okay the inscribed angle so the measure of angle a b c is going to equal half of the measure of the intercepted Arc half now usually I'm being asked to find the measure of inscribed angles not in this case I'm being asked to find the measure of the arc so why is this important well if the angle is half the arc then the arc is twice the angle that it creates so let me say that the measure of angle fgh I'm sorry the measure of Arc fgh which is U is going to be 2 times the angle created by these guys right here two times fah so the measure of angle f a h this is what I know this is what I know I know that I'm given information about this inscribed angle and that inscribed angle which means I care about the arc fgh obviously uh and I care about the angle or of the arc f a h obviously those two are connected and they both add up to 360 degrees so the measure of Arc f g h plus the measure of Arc fah is 360 degrees now I know the measure of angle fgh or the measure of Arc fgh is two times so you become two times the measure of angle f a h which means U would be two times the measure of angle f g h and that equals 360 degrees I know what angle fah is it's 38x Plus 5. I know what the measure of angle fgh is it's 21 x minus 2. and now it's just an ugly gross terrible Algebra 2 problem all right so distribute that become 76 plus 10 distribute again that becomes 42 x minus 4 and 360 is still 360. let's combine like terms let's combine like terms 46 or 76 x and 42 x is a hundred and eighteen X plus 10 minus 4 is plus six equals 360. subtract weird looking 6 subtract weird looking 6 118 x equals 354. divide both sides by 118 divide both sides by 118 and X is a perfect little three now why does this matter because originally what I was supposed to do is find the arc measure of fgh fgh is two times angle fah I now know that X is three so what I care about is two times 38x plus 5 were 38 times 3 is now being added to 5. 38 times 3 is 114. 114 plus 5 is 119 and twice that is 238 so the measure of Arc f g h is 238 degrees inscribed angles are usually pretty Pleasant this problem not so much determine if line a b is tangent to a circle what is a tangent line well let's say I give you a circle right here let's say that I put a center right there and I have a radius there if I were to have a tangent line a tangent line has to be perpendicular perpendicular to the radius wherever wherever that tangent line that hits the circle create the radius it has to be perpendicular which means you have to be a right angle right here which means in order for this to be a tangent line I have to show that this triangle is a right triangle well what information do I have well I have the fact that that radius is 12 which means this length right here is 12. I also know that this is 8 so this guy right here is 8 plus 12 which is 20. so I have a right triangle of 16 12 and 20. so I'm going to use the converse of the Pythagorean theorem to show that this is a right triangle so if I'm going to show that this is a right triangle I'm going to label u a and u b and I definitely have to label the longest side C and I have to make sure that a squared plus b squared is for sure c squared 12 squared plus 16 squared has to equal 20 squared 12 squared is 144. 16 squared 256. it's got to equal 400. 144 plus 256 is 400 does 400 equal 400 yes so is a b tangent to the circle yes again a tangent line has to be perpendicular to the radius that it hits so in other words since this is a right triangle it is perpendicular because it creates a 90 degree angle segment line indicated assume that the lines which appear to be tangent are a tangent and if you're wondering what's a tangent line a tangent line is a line that hits a uh Circle and is perpendicular to a radius it would be really be nice if I knew how to draw uh much better much better much better so what this means what this means is if this is a tangent line then you have to be perpendicular which means you have to be 90 degrees now I have to find this segment right here I'll call you x what do I know I know you were 1.5 and you were 1. so that means your 2.5 if you were my radius and the radius is 1.5 that means you're my radius and you're also 1.5 B so I have a right triangle where I care about finding a leg I'll call you a because you're a leg I'll call you B because you're a leg and I'll call you C because the hypotenuse has to be C doesn't matter which one's A or B so using Pythagorean theorem a squared plus b squared equals c squared I called a squared 1.5 so 1.5 squared plus b squared which is X so I'll leave it as B squared for now I could have called it x squared but too late I already wrote it 2.5 squared 1.5 times 1.5 is 2.25 plus b squared 2.5 times 2.5 is 6.25 subtract both sides by 2.25 subtract both sides by 2.25 and B squared is a really simple four is be all by itself nope square root it B is 2 and since I called BX X is 2. okay so tangents create 90 degree angles and usually if it gives you something that looks like a right triangle and it tells you if it looks like it's tangent it's tangent then you create some kind of 90 degree triangle looking thing Pythagorean theorem we did it we're done array find the perimeter of this polygon gross assume that the lines which appear to be tangent are tangent so your tangent your tangent your tangent your tangent Hmm this whole line is 18.4 a chunk of it is 5.4 this whole line is 21.9 this whole line is 20.5 I've got nothing about you this is what I know if it tells us that lines which appear to be tangent are tangent then we obviously care about tangents now this is what we know about tangents and about tangents that seem to meet at these points out here tangents that meet at a point in space are congruent which means you guys are going to be congruent you guys are going to be congruent you guys are going to be congruent and these two guys are going to be congruent so I know you're 5.4 now that doesn't give us this guy but we're going in the wrong direction if we try to go here we need to work our way around counterclockwise as gross as it is this whole thing is 18.4 if this chunk is 5.4 then you are going to be 18.4 minus 5.4 which is 13. so this guy is 13 which makes you 13 as well if this whole thing is 21.9 then I could subtract 13 from it to get this chunk right here 21.9 I'm sorry 21.9 minus 13 is 8.9 which means your 8.9 so if this whole thing is 20.5 and you are 8.9 then this chunk right here 20.5 minus 8.9 which is going to be 11.6 making you 11.6 now the reason why we had to go through that whole part of my language rigmarole is to find this whole length which is 5.4 plus 11.6 which is 17. so the perimeter of the quadrilateral is going to be 17 plus 18.4 plus 21.9 plus 20.5 okay all of the blue lines added together 17 plus 18.4 plus 21.9 plus 20.5 7.8 according to Mr computer that I'm using off to the side okay no craziness needed I mean it's a crazy problem it's a wacky problem but you know I'm very happy tangent's fun find the angle measure indicated assume that the lines which appear to be tangent are a tangent so there's only one line that appears to be tangent and it's U my goal is to find this fellow right here I know you're 44. so this is what I'm going to do I have a triangle living inside this triangle I don't know if they're similar they're probably not if they were similar then this would just be 90 degrees we call it a day but it's never that simple what I do see is this big triangle right here and I have two angles already for that big triangle I have this 90 degree angle I have that 44 degree angle and I have this guy which I don't know I'll call you X so what I know is that 90 degree angle there in green that 44 degree angle there off to the right and X are going to add up to make 180 degrees because of the triangle 90 plus 44 is 134 and now it's a one step problem equals 180. I'm going to subtract 134 from both sides minus 134 and X is going to end up being 46 whatever's degrees now why do I care about that why is that important well let's zoom out a little bit let's get rid of all this fluff if you're a radius that makes you a radius which makes you the same which makes you the same because of isosceles triangles oh so I'll call that question mark why and what I know now that if you're 46 degrees and you're 46 degrees 46 plus 46 plus I don't know which I called Y is going to equal 180 degrees because of triangles 46 and 46 are 92 92 plus y equals 180 degrees let's subtract 92 from both sides subtract 92 from both sides and it's going to end up equaling 88. so the missing angle measure that I care about let's put these back so it looks nicer for when I take a screenshot and make a a thumbnail for my YouTube account uh we're going to end up with 88 degrees oh fun isn't that nice it was super nice find the measure of the angle indicated what I have here is two chords now usually I play power chords on my guitar but two chords that's a different thing there's a little rule that says if you have two chords this also works for secants and you care about that guy and we'll call you a and we'll call you B and we'll call you C and we'll call you D there's a little rule that says if I want to find the measure of angle X here that what you do is the measure of angle X is going to be the average so half of Arc a b plus Arc CD okay so basically the we care about the arc that's created by these chords right we're not doing these guys we just care about the the I guess we want to say the intercepting arcs but I don't even know if that's technically true regardless regardless we're going to do the same thing here okay I'm going to call you X and so X is going to be the average of 175 and 55. so do I have enough space I might the measure of X so we'll just say x is going to equal one half of 175 Plus 55. which is going to be one half times 175 plus 55 is 230 which is when I usually go to the dentist and that ends up being 150. so the measurement that I care about the angle measurement that I care about question mark is going to be 115 degrees cords find the measure of the angle indicated these are secants right a secant is a line that basically has no you know Rhyme or Reason it's just a line that cuts through a circle all right now this secant by cutting it through the circle these two secants rather by cutting it through two circles gives me four points that it cuts through okay and we'll call them a b c and d now let's call you angle X which is kind of what I want to do here all right if I'm trying to find the measure of angle X which is an angle created by two secants what you do is the measure of angle X is going to equal one half of the large Arc that's created by the two sequence the large major arc so Arc BD minus Arc AC okay so close to the average but not quite all right so what I'm going to do is the same thing over here I need to find the measure of angle I don't know question mark we'll call you X so X is going to equal one half of the large Arc that's created so if I were to pretend that these lines are still going on we can see the same thing 175 would be that guy so 175 minus this smaller Arc that's created 75 57. oh numbers you're so difficult I'm not sure why I got two degrees in them 175 minus 57 I believe is 118. yep and and half of 118 is going to be 59. so X is 59 which means the measure of angle I don't know is going to be 59 degrees now okay fine secrets solve for x all right so I've got information here I've got information here and I have 50 degrees there's a little rule let me try to draw this rule as best as I can where if you take two secants okay two secants and that's essentially what this is if I wanted to I can keep these lines I can keep them going on so these guys now look a whole lot more the same and if I were to call you p q s and D to match up with this and call you R there's a rule that says if you take two secants and you have an angle created by those two secants intercepting the measure of that angle so the measure of angle R is going to equal one half of the larger Arc that's created so Arc PD in parentheses minus the smaller Arc that's created q s okay this problem just involves more math I did a problem like this before where it was just numbers now I have a little bit of like solving equations type stuff going on the measure of angle R according to this picture is fitty that's going to equal one-half I have to squeeze this in a little bit better PD is 12 X Plus 8. minus Qs now we have to be very careful we have to throw these in parentheses 4X plus 4 close it close it and now it's just a really really really ugly like equation that you would see I still have a bunch of stuff going on in parentheses so nothing happens with a half yet nothing happens with 12x yet nothing happens with 8 yet but what I can do is distribute this negative to make it minus 4X minus 4. that's why it's important to put those parentheses there okay inside the parentheses I can combine like terms so 50 doesn't change one half also doesn't change 12x minus 4X is 8x 8 minus 4 is 4. and now I distribute the half 50 equals half of 8X is 4X and half of 4 is 2. now it's a two stepper minus two minus 2 cross U out 48 equals 4X divide both sides by 4 divide both sides by four let's write it up here x equals 12. all right so secants cross to make an angle I'm going to take half of the difference of the two arcs that are created okay make sure it's positive the end yeah find the measure of angle QPR which is U now QPR is 26x minus 2. can't do much with that right now what I know is I have two chords or if you want to think about it two secants intersecting within a circle within the circle not outside the circle when you have two secants intersecting within the circle there's a little rule that I'll write up here okay and we'll call you a for angle we'll call Little a we'll call you a b c d the rule is that the measure of angle a is going to be the average of the intersecting Arc so the average of Arc a B plus Arc CD so that's how I'm going to set this problem up this angle is going to equal one-half times the quantity of 131 plus 18x plus 1. all right so let's set that up that's my angle right there I already have that information so 26x minus 2 is going to equal one half of one of the arcs so q r so one half of 131 that's a third one half of 131 plus 18x Plus 1. I can combine like terms within the parentheses so let me quick do that so one half of 132 blink-132 am I right plus 18x now we distribute that one half so 26x minus 2 is still there half of 132 is 66 plus half of 18x is 9x now I have variables on both sides of the equal sign let's get rid of the one on the right minus 9x minus 9x is going to give me 15x oh this is not going to be pleasant so 15x minus 2 is going to equal 66. I knew I was wrong this is 17x this is 17x 17x uh which now it's going to be pleasant because if I add 2 to both sides if I add 2 to both sides that gets me 17x equals 68 divide by 17 divide by 17 and that's going to get me x equals 4. done Circle it call it a day no because I need the measure of angle QPR and the measure of angle q p r is 26 x minus 2. I just found out that X is 4. so 26 times 4 minus 2. 26 times 4 is a hundred and four minus 2 is 102. so in the little space that I have down here on the bottom right I will write out that the measure of angle q p r is 102 degrees almost made a fatal mistake there find the angle measure indicated assume that lines which appear to be tangent are a tangent so let's take a look at this guy on the left you appear to be tangent so if that's a tangent line then what I know is U are 90 degrees this is basically an inscribed angle or better yet if this is 9 degrees this cuts the sky in half which means that right there is 180 am I done no because I still need this little sliver right here now this is 90 degrees as well if you were 76 that makes this little sliver 14 which makes you 28. so the question mark is going to be 180 plus 28 which is 208 degrees bada bing this guy looks bad but it's not so bad at all there's a rule that states that in this case the measure of angle U is going to equal the large Arc so I'll put half it's actually half of the large Arc so we'll put B for large plus this small Arc here's the issue I'll call you X because that's what I'm finding but I don't know what this is except for the fact that this whole circle is 360 and if your X then this Arc is going to be 360 minus X okay so what I'm going to do it's actually a minus I knew something was wrong there I was looking at that I was like I'm not doing it right it's a minus because the angle meets on the outside of the circle silly me minus all right so what I'm going to do is I'm going to say 50 because that is the measure of angle U is going to equal one half of the big guy which is 360 minus x minus the small guy which is X so 50 equals one-half of 360 minus 2X let's distribute that one half so 50 is going to equal 180 minus X If I subtract 180 from both sides which I'm doing right now I get Negative 130 on the left that's going to equal negative X and if Negative X is negative 130 then positive X is positive 130. which means the question mark for this one is 130 degrees solve for x assume that lines which appear tangent are a tangent so what we have is an intersection of a tangent line and a secant line there's a little rule that says if you have the intersection of a tangent line and a secant line what you do is we'll call this angle a okay I like to call this the far away Arc and I like to call this the near Arc the rule is the measure of angle a is going to equal one half of the far Arc minus the near Arc so the way that I can set this up the way that I can set this up is my angle measure I know to be 5X minus 5. that's going to equal one half of my far Arc which is 190. minus my near Arc which is 13 minus 7 in parentheses if you don't have parentheses things are going to go bad 13x minus seven double parentheses to end it out before I can do anything I have to distribute this negative so let me do that so now I have 5x minus 5 on the left equals one half of 190 minus 13x plus 7. this is not going to be fun at all not fun at all I'll show you a trick I'll show you a trick I'll show you my little trick one half times 190 plus 7 is now 197 minus 13x now that one half I don't want to distribute that one half so here's my trick If I multiply both sides by 2 If I multiply both sides by 2 then these cross out and all I have to do is distribute the 2 and everything works out 10x minus 10 is going to equal 197 minus 13x okay now I can add 13x to both sides which is what I'm going to do that gets me 26x I know 23x 23 x minus 10. equals 197. I'm running out of space so I'll write it over here all the way to the left 23x if I add 10 to both sides that gets me 207 divide both sides by 23 207 divided by 23 let me think 207 divided by 23 according to Siri so x equals 9. which is what I was supposed to do gross that looks awful well I'm having a little lag issues on my computer right now and I'm towards the end of making like a bunch of videos so I don't want to reset it now but that's how you do it X is nine not the prettiest problem in the world but it's done looks like a party hat doesn't look like a party hat it's a guy going to a party we're a bird maybe it looks like a bird solve for x all right I had two chords intersecting within a circle there's a little rule that says if you have two chords intersecting within a circle call the segments in one line a b called the segments in the other line CD and a b is going to equal CD so over here U would be like my a b so x times 4 is 4X and U would be my CD so 5 times 8 is 40. divide by four divide by four X is equal to 10. 0 funzo solve for x I have two chords intersecting within a circle there's a little rule that says if you have two chords intersecting within a circle call the two parts of one chord A B and call the other two parts of the other chord c d and a times B is going to equal c times D now what does that mean here I have to do a TNT bit of work for if you are my two my one chord connected four X plus two times eight is going to look like this so I'll do 8 times 4X plus 2 so it looks like distributive property it's prettier that way and then 9 times 4X 8 times 4X is 32x which is a video game system from back in the 90s 8 times 2 is 16. and 9 times 4X is 36x on both sides of the equal sign I want to keep it positive plus there's no number over here so it would be stupid for me to subtract 36x from both sides so I'll subtract 32x from both sides just like uh the country subtracted 36 x 32 forget it I'm just trying to make a joke uh that's going to equal 4X so 16 equals 4X divide both sides by 4 divide both sides by 4 x equals 4 and that's exactly what I'm supposed to do so there you have it X is four fun CE all right what do we have here we have two secants intersecting inside of a circle to make two chords and then stuff on the outside usually when we see two secants we're thinking oh this angle is going to be at one half of that Arc minus that Arc simple no we are finding measurements here there is a rule that says if you have two secants which hit a circle and create two chords and then the outer parts here what you do is you take the entire secant so I'm looking on this guy on the left the entire secant and multiply the outside part so s e times f e is going to equal the entire secant which is c e times d e so let's plug in what we have SE is going to be 10 plus 8 which is 18. I can do that f e is eight I can do that that's going to equal c e not so pleasant this one uh two x minus Seventeen plus nine so two x minus Seventeen plus nine I'll simplify that in a minute and d e is nine okay 18 times 8 18 times eight according to my watches that would be 144. thank you watch now I have 2X uh minus 17 plus 9 is minus eight so multiply that to nine so that's distributive property right there so 144 on the left is unfazed so I have uh 9 times 2x which is 18x 9 times negative 8 which is minus 72. now it's a two stepper I'm going to add 72 to both sides I'm going to add 72 to both sides and you become 216 that equals 18 x divide both sides by 18. and let me see let me see let me see I'm going to squeeze my watch again uh 216 divided by 18. that would be 12. so X is 12. am I done Gnar because my job is to find c e c e is 2x minus seven minus nine you were CE right there so c e is 2 times 12 minus eight two times twelve is twenty four twenty four minus eight is sixteen so c e is 16 no units that's my guy I am done identify the center and radius then sketch the graph well obviously if it's asking me to find radius I'm dealing with the circle here the formula for a circle is as follows x minus H squared plus y minus k squared equals r squared what's these H's what's these K's what's these R's you ask the center is going to be h k the radius is going to be r okay so let's find the center first centers h k negative H is negative two negative H is negative two which means positive H is positive two so if you're like oh does that mean negative K is positive one which means positive K is negative one yep now it says here that the radius r squared is 16. so if r squared is 16 I could just square root U and square root U and the radius is 4. so drawing these are always awful for me because of the pen and tablet thing that I use but you know we'll see what happens the center is right to down one that's my center right there my radius is four units away so I'm going to count one two three four to the right I'm going to count one two three four up I'm going to count one two three four left and I'm going to count one two three four down and I'm just going to try my best to make a circle try my best to make a circle try my best to make a circle try my best to make a circle you don't give me awards for drawing but they do give me awards for being a great teacher and that's what I try to do and fail but I try it's my picture I'm done write the equation of a circle if the ends of its diameter are 18 negative 3 and 4 negative three all right I'm going to sketch this out the best that I can alright without doing too much 18 negative 13 I think I said negative 3 18 negative 13 would be you know somewhere around here 4 negative 3 would be you know somewhere around here and what I know is these are the ends of a diameter which means circle is going to live somewhere around there perfectly drawn Circle you say I disagree the center of my circle is right there in the middle of the diameter that's important so thing number one that I need to do is find where the center of the circle is so let's do that let's get that out of the way right now the center of the circle is going to be the midpoint of my two endpoints of the diameter you find the midpoint by getting a point adding the x value so 18 plus 4 divide them by two adding the Y values so negative 13 minus 3 or plus negative 3 over 2 18 plus 4 is 22 Taylor Swift 22 divided by 2 is 11. negative 13 minus 3 is negative 16 negative 16 divided by 2 is negative 8. there's my center now I need my radius my radius is a length and my radius is going to be a line that connects an end point an end of the outside of the circle to the center so why don't I use the distance formula to help me find out the length of the radius the distance formula by the way this point right here the point that I'm using is going to be 4 negative 3 because the numbers are smaller so why not so I'm going to be using these two points to find my distance so you do One X value minus the other x value so 11 minus 4 it plus one y value negative 8 minus the other y value negative 3 and square that oh a double negative let's make you a positive so 11 minus 4 is 7 so 7 squared is 49. and uh negative 8 minus three or plus three is negative five and negative five squared is twenty-five okay so that ends up being that's not just not pleasant at all just not pleasant at all I'm just making sure my math is good and it is unfortunately it is um 49 plus 25 is 74. the square root of 74. now I'm going to leave that you'll see why because this is actually not super bad to quote a movie um the diameter is the square root of 74. the equation for a circle is x minus h squared plus y minus K squared equals r squared what's H and what's K my Center is H and K R means radius so the the distance that I found here that's my radius that's my R so I'm going to square root 74 nah I'll just cancel things out but we'll get there we'll get there okay we might have to do some cleanup so I'll just write out what I have x minus H is x minus 11 squared plus y minus K is y minus negative 8 squared we'll clean that up that's going to equal the square root of 74. squared so we'll make that a positive and the grand finale is going to be X minus 11. squared plus y plus eight squared equals these guys cancel each other out 74. um circles fun