Transcript for:
Geometry Basics Review

boys and girls this is mr moniz here with a super cool video in this video i'm going to walk you guys through the test review for our first unit in geometry unit number one on the basics of geometry i will make this disclaimer on all of my videos i love to make these videos for you guys so that you can go back and check these if you're not sure about a certain problem you can fast forward to the one that you need some help on but i need to make sure you guys understand these videos are not meant for you guys just to go home and watch these and copy down the answers and turn your new work into the box the next morning these videos are meant it's kind of a guide for you i really need you guys to work all these problems on your own hopefully we'll have time in class to finish some of this stuff but if you have to do some of this stuff at home i need you guys to figure out what you know and do as many of these problems as you can and then if you've got two or three that you're not sure about then come back to the videos and see if you can figure out where your mistakes were or to get that help that you need so without further ado let's go ahead and jump in here with our first problem on the first test review number one basics of geometry so in this first problem we are looking at different ratios for a line segment we're trying to find thirty percent of a line or five eighths of a line or some kind of ratio other than just the middle the midpoint of the line so we are partitioning a line and we have this kind of scary complicated formula here so i'm going to come over here to the side find another spot over here and we're going to write down our formula you guys will need to memorize these so when i'm trying to find a point of a line that we are trying to partition or ratio of a line our formula is going to be a times x one plus b times x two all over a plus b and that's going to be common that's going to be for the x coordinate and then the y coordinate is going to be a times y 1 plus b times y 2 all of that is also divided by a plus b so those will be the coordinates of our partition point now what are the a and the b the a and the b is simply the numerator and denominator of the ratio so if we go back to our problem over here we're looking at a ratio of four sevenths on this line so the numerator is going to be your a and your denominator is going to be your b okay so we're trying to find what is 4 7 or what is 4 7 of the way up this line okay i'm going to go ahead and note both of these points the first one that i see i'm going to let that be my x1 y1 so there's my x1 y1 there's my x2 y2 and i have everything i need i've got my a i have my b i've got all the x's and y's so i'm ready to simply plug it into our formula and it's going to look something like this our point is going to be equal to a times x1 plus b times x2 all right so i'm going to have 4 times negative 2 plus 7 times 1 all divided by a plus b i'm just going to add the numerator and the denominator together 4 plus 7 that's just going to be my x coordinate and then comma the y coordinate is going to be a times y one so four times three plus eight of b times y two so plus seven times seven all over a plus b four plus seven and then i need to go ahead and simplify this so that's gonna be equal to negative eight plus seven over eleven comma twelve plus forty over 11 and that is going to simplify to negative 1 11 comma 61 11. that is our final answer so if we were to take a ruler that has endpoints of these two points if i want to find four sevenths of the way up that ruler or that length then this these would be that would be the coordinate of that point all right number two is simply the same thing we've got another ratio in this case we're looking at two thirds my a is two my b is three that's my x one y one that is my x two y two and i simply plug it in i'm gonna go through this one a little bit faster so my point is gonna be equal to two times four and again the beauty of these videos guys is you can pause you can rewind i'm gonna put these on youtube so you can do the little back 10 seconds if you need to slow down or pause to catch up that's going to be my y-coordinate my my x-coordinate excuse me this is going to be my y-coordinate uh 2 times negative 2 plus 3 times five once again guys i'm just simply plugging it in to our partitioned line formula that's also two plus three that is going to be equal to eight plus 27 over five comma negative 4 plus 15 over 5 and so our point is going to be equal to 35 over 5 comma 11 over 5 and 35 over 5 simplifies so our final answer is actually p equals 7 comma 11 fifths seven comma eleven fifths okay so that's it for our two ratio problems kind of start off the test review with a couple of uh difficult problems but that's okay it's a little bit easier from here all right so now moving on number three find the value of m so our segment postulate says this guy a b plus bc equals the whole length ac in this case they're giving us ticks here so i know that b is the midpoint so that means that a b is the same length as bc find the value of m these two guys are the same length so i simply set these guys equal to each other 3m plus 5 equals 4m minus 10. that's super easy subtract 3m from both sides 5 equals m minus 10 and add 10 to both sides and i get m is equal to 15. okay simple enough oops sorry make sure i always have to check make sure you guys can sorry see what i'm doing in the camera i'm writing on the paper but i always need to make sure that you guys are seeing what i'm doing here all right so let's check this i can always go back and check my answer make sure i did this correctly so if i stick in a 15 here 3 times 15 is 45 plus 5 a b is 50 4 times 15 is 60 minus 10 that's also 50. so guys please take those couple extra seconds and make sure you're going back and checking your work to make sure you don't make a careless mistake all right moving on to the next one here and you guys will have to kind of forgive my camera sometimes sometimes it goes out of focus and i have to kind of shake the paper to wake it up make sure it gets back into focus here come on you can do it sometimes there we go thank you welcome all right so x z the whole length is 50. what are x y and y z so if the whole length is 50 i'm going to say this guy x y plus y z is equal to 50. super easy so i'm going to say a plus a plus 8 all that's going to be equal to 50. so i have 2a plus 8 equals 50. subtract 8 from both sides 2a equals 42 divide by 2. about 2 and we get a is equal to 21. okay i can check that if this length is 21 this is 21 plus 8 that's 29 does 21 plus 29 does that equal 50 yes it does okay so once again guys you can tell it doesn't take but just a second or two to go back and check these make sure you do these correctly all right let's move on move on all right so on the next page numbers five and six numbers five and six let's get these in focus here for you all right so we'll call this number five we'll call this number six okay so we're going to use the diagram for both of these guys it says if the measure of angle mqr is 61. so this guy right here mqr so if that guy is 61 okay the measure of mqp so this guy this guy is 25 degrees find the measure of angle pqr okay so we want this guy right here pqr okay so our angle addition postulate says if i add this little angle plus this angle that will equal the whole angle so i'm trying to find this missing angle here p q r so i'm simply going to subtract the big angle minus the little angle that is going to be 61 minus 25 and that gives us 36 degrees 61 minus 25 36 degrees okay number six if the measure of angle nqm okay so this guy right here n q m if that guy is two x plus eight okay two x plus eight and the measure of angle pqr okay this guy right here this guy if that is x plus 22 find the value of x well if you'll notice guys in the diagram we have these little arcs right so if they give us the little tick marks if they give us little arcs that means that those angles are identical so 2x plus 8 and x plus 22 will have the same measure so i'm simply going to set those guys equal to each other so i have 2x plus 8 is equal to x plus 22. subtract x from both sides x plus 8 equals 22 subtract 8 and i get x is equal to 14. okay zoom out just a little bit more so i keep having to shift my camera so much okay let's move along all right number seven says find the distance between the points to the nearest tenth now we don't have a coordinate grid here so again if you guys want to use pythagorean theorem you're welcome to um if it says use the distance formula then you do have to use the distance formula i didn't say that here just says find the length find the distance but we don't have a coordinate grid so probably in this case it is easier to use the distance formula so let's go ahead and write that down our distance formula is simply the difference in our x's so x2 minus x1 i'm going to square that plus the differences of my y's y2 minus y1 and then i'm gonna square that so once again guys it does not matter which one you label x1 or x1 or x2 because we're going to end up squaring the difference so it doesn't really matter i'll let these be my ones x one y one x two y two okay so i'm simply going to plug and chug plug it into our formula distance is the square root of that guy minus that guy but remember that's a double negative it's x2 minus x1 but x1 is a negative so you got to make sure you put that in parentheses use those double sets of parentheses so you don't make a mistake plus y2 4 minus y1 4 minus 5 i'm going to square that okay so zero minus a negative one that's zero plus one uh so one squared is one and then four minus five is negative one negative one squared is also one so it's one plus one which gives us the square root of 2. now notice in the directions it says nearest tenth so on your calculator you're simply going to plug that in so on your calculator now the square root button is right here where my thumb is above the x square button in blue so i'm going to press second square root and then 2 press enter your calculator will spit out the decimal rounded to the nearest tenth is approximately approximately 1.4 okay number eight we've got basically the same thing so x1 y1 x2 y2 so distance is going to be the square root of that guy minus that guy once again we've got a we have a double negative we have 6 minus a negative 1 squared plus 2 minus a negative 1 squared okay so that is going to give me the square root that's going to be 6 plus 1 which is 7 7 squared is 49 2 plus 1 is 3 3 squared is 9 so we have square root of 49 plus 9 which is the square root of 58 and that is approximately 7.6 okay a distance formula pretty easy just make sure you guys don't get tripped up here with these double negatives all right for number nine and 10 they give us this here a b has end points a and b and in number nine it says find the coordinates of the midpoint okay so the midpoint remember the midpoint is the middle so you need to think average right in the center okay so my midpoint formula if we want to write that down here the midpoint is equal to it's the average of the x's so whenever i want to find the average i add them together and divide by two so i add my x's together x one plus x two divided by two and then add my y's together and divide by two again it's just the average of the x's comma average of the y's so in this case if i plug this in we can let this be x one y one x2 y2 if i plug that in i'm going to add my x's together so i have negative 3 plus 3 divided by 2 comma 2 minus 2 divided by 2. and that comes out to be zero over two comma zero over two and guess what that is the origin folks so the midpoint of that line is right smack dab in the middle of the coordinate plane zero comma zero okay number 10 is we're going to take these same points this is find the length of a b to the nearest tenth so once again we don't have a coordinate grid so i'll use the distance formula here so distance equals the square root of we've got three minus a negative three squared plus negative two minus two squared that's going to become that's three plus 3 which is 6 6 squared is 36 and then negative 2 minus 2 is negative 4 negative 4 squared is 16. add those together and i get the square root of 52 which is approximately 7.2 okay number 11 number 11. okay so 11 and 12 are a little bit tricky these are the ones that students tend to get hung up on just a little bit now normally we're trying to find the midpoint but 11 to 12 that give us the midpoint we're trying to find one of the other ends of the segments okay so i have a segment here they give us that check they give us the midpoint check we have to find the other end of the segment okay so we gotta work backwards so they tell us that the midpoint is negative one one okay so in a midpoint formula i'm going to come out here to the side and say okay negative one comma one that midpoint is equal to okay i'm going to set that equal to the point so we have this point so i'll call this i'll call this x1 y1 okay so i have this x coordinate negative eight plus the x other x coordinate i don't know what that other x coordinate is okay so i'm just gonna put x two divided by two okay comma my y-coordinate okay of point j which is four plus the other y-coordinate y'all give me just one second hold just one second [Music] right [Music] okay sorry about that guys my dog was at the bottom of the stairs and he was uh he was whining for me okay sorry about that okay so we've got the y coordinate of our j plus the other y coordinate which we don't know so i'm going to say y two divided by two okay now we don't know what x2 and y2 are but i know that all of this stuff right here this guy is equal to the x coordinate of the midpoint this guy is equal to the y coordinate of the midpoint so i'm simply going to set all this stuff equal to negative 1. so i have negative 8 plus x 2 divided by 2. i'm going to set that equal to the x coordinate of the midpoint okay i got this 2 on the bottom so i'm going to clear that so i'm going to multiply both sides by 2. those twos will cancel so i have negative 8 plus x 2 is equal to negative 1 times 2 which is negative 2. i'm going to add 8 to both sides and i get x 2 is equal to 6. okay now i've got to come back over here to the y's and i'm going to set all this stuff equal to the y coordinate of the midpoint so i'm going to say 4 plus y 2 over 2 is equal to that guy which is 1. once again i'm going to multiply both sides by 2 so i clear that clear that fraction that i don't like so i'm going to have 4 plus y 2 is equal to 2. i'm going to subtract 4 from both sides and i get y 2 is equal to negative 2. so the answer to this question the coordinate the other end of my segment this guy right here that we're trying to find the coordinates are 6 comma negative 2 that is point k okay that is the other end of this segment okay all right we're going to do the same thing over here for number 12. once again m is my midpoint so i'm going to set that guy equal to 5 comma negative 2 is going to be equal to here's my x1 y1 that's going to be equal to the average of the x's so it's going to be the 9 plus the other x that i don't know divided by 2 comma the y that i know negative 5 plus the y that i don't know divided by two so i'm going to set all this equal to the five i'm going to set the y coordinates equal to the y coordinate of the midpoint so for the x i'm going to have nine plus x two over two is equal to the five i'm going to multiply both sides by two see you later see you later so i have nine plus x two is equal to ten subtract nine and i get x two is equal to 1 and then i'm going to set all this equal to the y coordinate of the midpoint so i have negative 5 plus y 2 over 2 is equal to that guy negative 2 let's multiply everything by two two's cancel make sure y'all can see that negative five plus y two equals negative four let's add five to both sides and i get y two is equal to one so my final coordinate point k is going to be equal to that guy and that guy one comma one and that is the other end of the segment one comma one all right moving along here number 13 we're trying to find the perimeter and the area of each figure this is super easy this is a square so of course all sides are length of eight the perimeter we're just going to add those up so the perimeter is eight plus eight plus eight let's say or simply four times eight which of course is thirty two now remember perimeter we're in one dimension land okay we're just adding up a whole bunch of links so the units are gonna be centimeters miles feet one dimension okay now the area is two dimensions so the area is going to be the side squared or eight times eight or eight squared 64 but now we're in two dimension land so this is going to be centimeters squared okay all right number 14 we got kind of an odd figure here to find the perimeter i need to use all these tick marks so everywhere i see a double tick that means that's going to be five inches so that's a five that's a five that's a five everywhere i see a single tick that's going to be three so that's a three that's a three that's a three this is a three okay so to find the perimeter i gotta add all these guys up so my perimeter is gonna be five plus five plus five plus three plus three plus five plus three plus three five five five i believe we have four fives and we have four thirties and that comes out to be 32 inches once again we're in one dimension land okay now for the area to find the area of all this stuff in here is that not just the area of the square plus the area of the rectangle because this is basically if we just draw an invisible line down through through there it's basically just adding a square plus a rectangle so the area is going to be the area of the square plus the area of the rectangle and the area of the square is just 5 squared plus the area of the rectangle is going to be the length times the width 3 times 5. that's going to be 25 plus oops sorry 3 times 5 25 plus 15. and so the area is going to be equal to 40. we're in two dimension land so that's going to be inches squared okay again if you guys think i'm going too fast pause it rewind it 10 seconds slow it down and make sure you're comfortable with everything we're doing here okay number 15 circumference and area okay find the circumference so the formula for circumference is pi d or 2 pi r the area of a circle is pi r squared so in this first one they give us the radius so i'm going to use this guy so for the circumference is 2 pi r 2 pi times 3 which is 6 pi and i'll notice guys they say leave it in terms of pi so when you see that do not type in pi in your calculator your final answer will have literally have the symbol pi in it and you just multiply everything else together circumference remember that's the outside of the circle that's the perimeter so we're still in one dimension land okay so that's just six pi inches for the area that's pi r squared which is pi times three squared which is nine pi now now we're in two dimensional and because it's area so this is inches squared or square inches okay 16. now they give us the diameter so i'll just use this guy circumference is pi d so it's literally just 15 pi meters because we're in one dimension land the area is going to be pi r squared well if the diameter is 15 then the radius is half of that right 15 halves or 7.5 so the area is going to be pi times 15 halves squared so if you keep it as a decimal that comes out to be 56.25 pi that's meters squared if you decided to leave this as a fraction and square top square the bottom that comes out to be 225 over 4 pi meters squared if it doesn't say i don't really care how you how you guys would leave that on a test or quiz that would be up to you okay so that's circumference and area of a circle all right moving along making your progress here all right on over to number 17 and 18. let me zoom out just a little bit here okay so we're using this figure figure for 17 and 18. okay so it says number 17 it says now notice here guys notice this little box that is a right angle that is 90 degrees and then notice we also have these little arcs so those two angles are the same so this first one says the measure of bdk okay this the small angle here bdk is 3x plus 4 and then the measure of jdr the other small angle is 5x minus 10 but we just saw that those two arcs are there so that lets us know both of these angles are the same so i'm simply going to set these guys equal to each other also remember guys if this were if this were two arcs then we couldn't do that remember it's got to be the same number of arcs one arc here one arc here they're identical okay so we solve this you guys can just do this in your head i'm sure 4 equals 2x minus 10 add 10 to both sides 2x equals 14 divide by 2 and we get x is equal to 7. now is that what they asked for sure is find the value of the variable that's what we did again for the sake of this video i'm not going to go back and check everything but of course you guys can plug this back into both of those make sure that they do indeed equal each other all right number 18 the measure of bdj okay so this guy here bdj okay so this intermediate angle is 7y plus 2 and jdr jdr the rest of it is 2y plus 7 okay so they give us this angle and then they give us this angle okay well the intermediate angle plus the small angle equal the whole angle right well we just noticed at the very beginning that that's a box right that's 90 degrees so both of these angles have to add up to 90 degrees okay these are complementary angles you guys have probably heard those terms before supplementary and complementary complementary as to 90 supplementary means to add up to 180. so i'm simply going to add these two guys together and set it equal to 90. so 7y plus 2 plus 2y plus 7. that's going to equal 90 degrees so combining my like terms 9y plus 9 equals 90. subtract 9 from both sides 9y equals 81 divided by 9 and we get y is equal to nine okay again i could go back and plug this in and make sure those two angles add up to 90 degrees okay moving along 19 20 21 24 they're asking are these always sometimes never true t q and q t it doesn't matter which way you notate a line so yeah that's always true j k and j l are the same ray that's going to happen sometimes it could be they could lie on the same line so if i had a ray if i had j k and l yeah they're on the same ray but i could also have this i could have ray j k and a different ray j l so sometimes they're the same ray intersecting lines or coplanar yeah that is always true two lines that intersect that will always be in a two dimensional plane four points are coplanar that sometimes they might be if i had a plane they might three you know three of these points could be on the same plane but this fourth point could be in a totally different plane um that's not in the same two-dimensional space so they could lie on the same plane but they don't have to a plane containing two points of a line contains the entire line okay that's that's always true two a plane containing two points of a line contains the entire line so if i have a plane and that contains a line this whole plane i'm sorry this whole plane will always contain this whole line that extends in either direction uh 24 two distinct lines intersect in more than one point that's false that that can never happen two two lines can never intersect in more than one point otherwise they'd be curvy crooked all right number 25 we're going to use this diagram over here for number 25 and 26. so 25 says if a d okay if a d if that guy is 12 okay and ac the whole length if this guy that whole length is 4 y minus 36 okay then find the value of y well we've got double ticks here right so i know that guy's 12 i don't know that guy is 12. so if i double that both of those guys are 12 so if i multiply that times two or if i just add them together i can set that equal to the whole length so let's just find the value of y but then it also says find the value of ac and dc okay ac dc i love that band by the way back in black our band used to play that song so again once once again i got two of these guys so i could say 2 times 12. make sure i can see that 2 times 12 okay has got to equal the whole length which they told us was 4y minus 36 okay so i got 24 equals 4y minus 36. i'm going to add 36 to both sides 4y is equal to 60 divided by 4 and we get y is equal to 15. okay so we found y and now we got to go back and find ac and dc okay so dc we already know it's 12 because if that piece is 12 and that piece is 12 so that's easy so dc is just another 12. and if i want to find the whole length ac i just simply plug in uh plug in 5 of 15 for y here so i have well actually there's two ways to do it okay if i know this is 12 and i know this is 12 then i simply just add them together and i can say that ac is 12 plus 12 which is 24 or you could have done it a long way and plugged in a 15 here 4 times 15 minus 36 and you have also gotten 24. so that is the length of ac i love math lots of different ways to get to the same answer okay number 26 we're using the same diagram here number 26 says if e d is x plus 4 and db is 3x minus 8 find ed db and eb sounds like red runner okay ed let's take this one at a time e d so this guy is x plus four okay e d is x plus four and db the other side is three x minus eight all right i got a single tick and a single tick so that lets me know that each of those pieces are the same so i can set these two expressions equal to each other and solve so i can say that x plus four equals three x minus eight let's move the x to the other side 4 equals 2x minus 8. let's add 8 to both sides 2x equals 12. so x equals 6. okay so we got x that's pretty easy and now we've got to go back and find e d okay so five e d i just sticking a six right here so e d is simply six plus four just sticking a 6 right there which is 10 okay i got to find db well if ed is 10 then db is the same as zd so that also equals 10. and that says eb the whole length well i'm no math teacher but i believe eb is simply the sum of both of those together and i believe that's 10 plus 10 which is 20. super easy stuff okay all right let's keep going 27 if the measure of angle abd okay so the whole angle is 79 what are the measures of angle abc the intermediate one and the small one d bc again our angle postulate says angle sum postulate says if i add these two angles together they will equal the whole angle so i'm simply going to add these two expressions together and set it equal to 79 degrees that's a little degree symbol by the way that's not a squared that's kind of hard to see so i'm going to set 5 x plus 4 plus 8 x is that a minus that is a minus 8x minus 3. then i'm going to set that equal to 79. and let's combine the like terms so i've got 13x plus 1 equals 79 13x equals 78 move the one to the other side divide by 13 i get x is equal to six so once again we get another six okay and then it says go back and find the measures of each of these angles okay so i'm simply going to take this plug it back in here take this plug it back in here so i can say that the measure of angle abc this guy right here is going to be eight x plus three or eight times six plus three just i keep saying plus that's a minus and that's going to be 48 minus 3 which is 45 degrees and then the measure of angle dbc is going to be equal to plugging it in over here it's going to be 5 times my x which is 6 plus 4 30 plus 4 gives me 34 degrees now how can i check this we said that the sum of both of these angles had to be equal to what right it's got to be equal to 79 so 45 plus 34 does that equal 79 yes it does that took me three seconds to go back and check and make sure i didn't make a careless mistake anywhere so please do that guys 28 okay so we have a straight angle we know that the sum the measure of a straight angle is 180 degrees it's half a circle right so it says what are the measures of angle rqs and the measure of angle tqs so i'm simply going to add these two guys together and they have to equal half a circle so i'm going to say that 6x plus 20 plus the other angle 2x plus 4 all of that has to add up to 180 combine my like terms i have 8x plus 24 equals 180 move the 24 to the other side 8x equals a 156 divided by 8 and x equals 19.5 now when you get 19.5 that ought to be a red flag that says oh wait a minute normally these kinds of problems i normally get a whole number now the fact that i got a 19.5 doesn't scare me too much but if i've got 19.6238 i don't thought um i might have made a mistake somewhere so you might want to go back and double check that hopefully you guys did that if you went back and double checked you would have seen that you know what it is 19.5 that is correct okay so now i got to go back and find what did my boss ask me to do i need to find the measure of these two angles okay so once again i'm just going to plug it back into each of these expressions to figure out the angles so the measure of angle rqs which is this guy over here on the right that is going to be equal to 2 times my x 2 times 19 and a half plus four that comes out to be 43 degrees and then the measure of the angle on the left the measure of angle t q s is equal to six times the 19.5 plus 20 and that comes out to be 137 degrees so once again does 43 plus 137 does that equal 180 you betcha so i feel pretty good about my answer alrighty let's go let's keep going all right so all right number 29 and number 30. we have been asked to graph each figure and find the perimeter of both of these guys okay so in number 29 i'm going to plot each of these points first on the coordinate grid so i have negative 4 1 2 3 4 left 1 down 1 that's my a b i have right 4 1 2 three four up one two three four five that's my b and c i have four down two that's my c i don't have a ruler with me at home so i'm very sorry that these are not perfectly straight lines that's not too bad and i need a perimeter okay so remember you don't need the distance formula or pythagorean theorem if it's horizontal or vertical line you just simply count or you just look at the difference of the x or y coordinates depending on which line it is so the length of bc i can count with my fingers one two three four five six seven or i could just look at the y coordinates of b and c i've got a five and a negative two what is the difference between five and negative two that is a length of seven okay now for ac and for a b again it doesn't say what to do in the directions i'm okay if you guys want to use pythagorean theorem i will actually do both just so you can see how this works for ac i'm actually going to use the distance formula because that would be kind of a small right triangle anyway so for ac i'll use the distance formula so i'll say a's distance for ac is going to equal to the square root of again i like to label these things x2 y2 so we'll go x2 minus x1 so that's 4 minus a negative 4. i love to throw those double negatives at us plus the differences in our y's negative 2 minus a negative 1 squared that's the square root that's 4 plus 4 that's 8 8 squared is 64. negative 2 that's a plus 1 that's negative 1 negative 1 squared is 1. okay so that comes out to be the square root of 65 which is approximately 8.1 okay now for a b i'm actually going to use pythagorean theorem so i'm going to i'm going to forget about this i'm just going to take this line and i'm going to form a right triangle out of this so that a b would be the hypotenuse of my right triangle so i'll do this kind of in dashed lines hopefully y'all can see that i'm just i'm forming a right triangle right here okay and then i'm going to use pythagorean theorem so this distance is one two three four five six seven eight that's a distance of eight this length is one two three four five six that's a length of six so i just do a squared plus b squared equals c squared a squared plus b squared equals c squared okay so my a it doesn't again it doesn't matter which one is your a and your b so it's a squared plus b squared equals c squared over here that's 36 plus 64 equals c squared c squared equals 100 take the square root of both sides and c equals 10. so to me guys i can do that much faster than i can do the distance formula so again it's your call but it says find the perimeter okay so we got to add all these up so the perimeter is going to be that the 10 plus the 8.1 okay plus the 7 over here so we said this length was 10 this length was 8.1 okay so we just add those three up and we get a perimeter of about 25.1 okay number 30 we have a quadrilateral so let's go ahead and plot those points 0 comma 1 is right there that's l we have 3 5 1 2 3 1 2 3 4 5. that's m n is a couple points over and p is four units below that so basically we have a right trapezoid that is p okay so three of the lines are horizontal vertical so we can just count that's a length of two that's a length of four this is a length of five so basically we just need to find the length of lm so once again you could use pythagorean theorem i'll do distance formula on this one even though i could probably do pythagorean theorem a little bit quicker okay so the distance of lm x1 y1 x2 y2 that is going to be the square root of 3 minus 0 squared plus 5 minus 1 squared that's going to be square root of 3 squared plus 4 squared 9 plus 16 square root of 25 which is 5. all right so my perimeter in this case is pretty easy we found lm so we just add these up five plus five plus four plus two you can't see that five plus five plus four plus two and we get 16 [Music] already all right now we're on to the word part of geometry in this first unit conditional biconditional converse statements 31 says identify the hypothesis and conclusion remember hypothesis is what comes after the if that's the front part of the conditional the conclusion is the back end it comes after then so the hypothesis in this case is the front-end hypothesis is you are an american citizen and i can reasonably conclude that if that is true you have the right to vote by the way is that true it's actually not true right seven-year-old can't vote even though they may be an american citizen but we understand what they mean okay 32 write the following is a conditional statement remember conditional has an if and event in it a counter example shows that a conjecture is false so if it is a counter example or if something is a counter example comma then it shows that that bad boy is false so the conditional would look something like this if something if a word if a phrase is a counter example i have to have this magic word then it shows that a conjecture which is like a prediction conjecture is a false okay number 33 determine if the conditional is true false write the converse determine if it's true false if it's false find a counter example okay so number 33 if a polygon has eight sides then it's an octagon is that true you betcha and that's true all right the converse remember the converse i'm just going to flip that around if a shape is an octagon then it has eight sides okay is that true it's an octagon that has eight sides yeah you betcha that's true so there is no counter example because both of those are true number 34 if you live in a country that borders the united states then you live in canada that bad boy is false what is a counter example mexico right mexico mexico i love mexican food mexico is a counter example of that statement if we write the converse however if i flip that around and say if you live in canada then you live in a country that borders the u.s that is true if you live in canada you live beside the us all right number 35 right the following is a conditional term for this truth by the converse determine that's true if they are both true then write by conditional if the converse is not true give a counter example two lines that lie in the same plane are coplanar yeah if they lie on the same plane they're coplanar that is the definition all right so if i flip that around the converse okay if two lines are coplanar then they lie in the same plane okay is that true yeah if they're coplanar by the way this is a good definition because this literally is the definition okay so yeah both of those are true all right so if i want to write this as a bi-conditional remember i have to have that magical phrase in the middle if and only if so i can say now remember if when i write the by conditional i take out the if i take out the then i throw an if and only if in the middle so my by conditional statement i'm abbreviating here two lines are coplanar if and only if comma they lie in the same plane okay all right let's keep moving along moving along okay again for sake of video i don't i typically try not to let these videos get too long so for the sake of the video i'm just gonna show you guys uh what the answers are for these next two it's basically the same thing write the conditional right it's converse if the converse is true combine them as a biconditional um this first one here if two segments have the same length and they're congruent if i flip that if two segments are congruent then they have the same length so i'll just put my answer key up here for these y'all can check these on mine okay so if two segments are congruent then they have the same length that's true so i can combine them in a bi-conditional and say two segments are congruent if and only if they have the same length okay now for this one for 37 if a number is divisible by 20 then it's even the converse if it is even then is divisible by 20 that's absolutely false okay there are lots of numbers um lots of numbers are even they're not divisible by 20 of course okay wrapping it all up here last couple of ones we can do it made it all right 38 now is when we start to increase the rigor okay so that's the kind of stuff you see on a regular geometry test you guys are in the honors class so this is the kind of stuff that you guys will have to do so we're going to be in quadratic land here so number 38 is asking us to find the length of lm mn and ln okay so let's go ahead and label this stuff so lm 2x squared plus 3x minus one and then m n is three x squared plus two x minus thirteen okay and the whole length this whole guy is eight x squared minus 9x minus 19. okay so i got to set this guy plus this guy equals the whole length so here we go you can do it i'm going to add these together so i have 2x squared y'all can see this 2x squared plus 3x minus 1 plus 3x squared plus 2x minus 13. that equals the whole piece which is 8x squared minus 9x minus nineteen combine my like terms five x squared plus five x minus fourteen equals the whole piece eight x squared minus nine x minus nineteen okay again combine all like terms let's move the 5x squared to the other side remember combining like terms you have to have the same variable and the same exponent so i have 5x minus 14 equals 3x squared minus 9x minus 19. i need my quadratic set equal to 0 right so i need to move both of these guys to the other side so i'm subtracting 5x and i'm adding 14. subtracting 5x adding 14. those go away so now i'm just going to flip this around so now my quadratic is 3x squared minus 14x minus 5 and now it's set equal to 0. okay so how do i solve this quadratic okay i can't use the gcf method because there's a c term negative 5 is not a perfect square so i can't use special case i have a b term so i can't do the square root method obviously a is not 1. so here's where i have to factor by grouping okay i gotta do the rainbow thing right three times negative five is negative fifteen i'm looking for factors of negative fifteen that will add to this middle term negative fourteen that should be pretty easy fifteen and one but which one has to be negative remember the sign of the middle term tells you the sign of the bigger number okay so my two factors are 15 and 1 but the 15 is going to have to be negative so i've got a negative 15 and a positive 1. sorry that's hard to see negative 15 and a positive one so i'm simply going to split this middle term i'm going to split negative 14 into negative 15 and one dollar okay and then i just bring down the rest of my stuff okay i still have negative 14x i just split it why did i do that the whole purpose of doing that is so that i can factor by grouping with four terms i draw my little wall my little curtain down the middle and i'm looking what is the gcf what's the greatest common factor i can pull out of both of those terms i can pull out a 3x that will leave me with x minus 5. what can i pull out of that stuff nothing right they don't have anything in common so all i can do is pull out one and that will leave me with x minus five okay these two guys match that makes me feel good okay so one of my factors is the x minus five the other factor is the stuff left over on the outside three x plus one okay so i'm gonna bring this back up over here i have some more room okay so now i gotta i gotta solve this right okay i can flip you flip over here because that coefficient is one so i know that one of my x values is five this one i gotta set equal to zero so if i move the one to the other side that's negative one then divide by three my other solution will be negative one third but here's the deal you gotta go back and take each of these guys and plug it back into plug it back into your original problem and see if it works out okay so if i take this negative one-third i'm going to plug it into this guy right here now i want you guys to do this with me on your calculator if you're still able if you're still keeping up and watching the video here's how you do this i'm going to take the negative one third i'm going to stick it in for x right there so i have two parentheses and make sure you pay attention to the negative button that's the subtraction button that's the negative button okay so don't get those confused so 2 times negative 1 divided by 3 okay i'm going to square that okay squared and then plus 3 parentheses negative 1 divided by three close the parenthesis and then minus one and what do i get i get negative one point eight so basically what we're saying is when i plug this back in here this length is a negative one point eight i cannot have that okay so this solution to our quadratic most the time we have two solutions that solution does not work so the only solution that does work is five okay but then it asks us to find l m m n and l in okay so i'm going to take the five and i'm going to plug it back into each one of these guys okay i'm just going to do one of these again for the sake of the video i'll plug it back in and find ln so if you guys want to do this with me i'm going to plug it into the bottom one so i have 8 times 5 squared minus 9 times 5 minus 19 and i get 136. so ln is 136. again for second time i'm not going to do this if you plug it back in here then lm comes out to be [Music] 72 and if you plug it into mn that comes out to be 64. then when you add those guys together 72 plus 64 you do add up to the 136. so that basically guarantees me that i did the problem correctly okay all right last couple of ones and we're done okay 39 all right so now we're going to use this diagram up here so 39 they give us this angle come on camera you can do it i need you to get back him i need you to get back focus for me you can do it sorry guys sometimes my camera doesn't want to uh sometimes i have to kind of take it away and bring it back there we go now we're back in focus okay so they're asking us to find the measure of x y t so that asks us to find this guy right here in this first problem okay so x y okay so x y t is three fourths x minus six z y t the rest of it is two thirds x plus eight and then x y z the whole angle is two x minus thirty three so i've gotta take this angle plus this angle and that's to equal the whole angle okay so here i'm going to add 3 4 x minus 6 plus i'm adding these two angles plus 2 3 x plus eight and that's going to be equal to the whole angle which is 2x minus 33. okay so here's where we refreshed our memory on how to solve equations with fractions first of all i'm going to go ahead and combine my like terms so i've got three-fourths x plus two-thirds oops two-thirds x and that becomes plus two equals two x minus thirty-three i gotta get a common denominator the common denominator of 3 and 4 is 12. so i'm going to multiply both sides by 12. 12 times 3 4 x is 9x you gotta distribute here 12 times two-thirds x is 24 divided by three that's what's 8x and then 12 times 2 is plus 24 12 times 2x is 24x 12 times negative 33 is negative 396. don't let that big number scare you combine my like terms 17x plus 24 equals 24x minus 396. let's move the 17x over here so i have 24 equals 7x minus 396. add 396 to both sides i'm going to come back up here i have more room 7x is equal to 420 divided by 7 and x is 60. again i'm kind of going through that fast just slow down the video pause it if you need to and then it asks us to find the measure of x y t okay so i'm simply going to take this and plug this back into x y t and then that's going to be so the measure of angle x y t is going to be 3 4 times 60 minus 6 and that comes out to be 39 degrees okay last one guys and it's another quadratic so they're telling us that x y t this guy is three x squared plus five x plus seven they're saying that t y z the rest of the angle is x squared minus two x plus six and they're telling us that x y z the whole angle the whole angle is 58 okay 58 degrees okay so i'm going to simply add these two guys together and set it equal to 58 so if i do that i have 3x squared plus 5x plus 7 plus the other angle x squared minus 2x plus 6. all that's going to be equal to 58. combine my like terms i have 4x squared plus 3x plus 13 equals 58. i need my quadratic set equal to zero so i'm going to subtract 58 from both sides 4x squared plus 3x minus 45 equals 0. okay can i pull out a gcf i always want to see if i can pull out a gcf i can't because i've got the 4 and 3 there so i can't pull anything out so now again i've got to do factoring and trinomial when a is not one so i can do the rainbow thing 4 times negative 45 it's a big number that comes out to be negative 180 i got to find factors of negative 180 to add to three so in other words i'm trying to find two numbers that multiply to give me 180 there are three apart from each other that's a big number i don't expect you guys to be able to do that in your head so you got to use a little trick on the calculator that i taught you so what you're going to do is you're going to press this y equals button y equals and then you're going to type in the number divided by x so 180 divided by and then your x button is right here my finger is 180 divided by x and then you're going to go second table where my finger is right here second table this will give you a table of values these are all the factors of 180 so i'm essentially going down through this list and trying to find which pair of numbers are three apart from each other and there they are 12 and 15. they're three apart one of these guys since the c is negative one of these guys has to be negative one of these guys has to be positive okay when i say that they're three apart that little trick only works when the c is negative okay so one of these guys has to be positive one of these guys has to be negative the middle sign is positive so the 15 is positive 12 is negative so i'm going to split this guy i got 4x squared it doesn't matter which one i put first i'll put the 12 with the 4 they're both divisible by 3 that kind of makes sense to me so i'll do 4x squared minus 12x plus 15x minus 45 equals 0. i can draw my curtain what's the gcf of 4x squared and 12x that's 4x that will leave me with x minus 3. what is the gcf of 15x squared negative 45 that's simply a 15 that will also leave me with x minus 3. that makes me feel good so these guys match so one of my factors is x minus three my other factor is four x plus fifteen now here i can flip you flip so x is equal to three here i move the 15 to the other side negative 15 divided by four my other solution is negative 15 fourths and when you first look at this you're thinking okay there's a negative x i'm sure that's probably the one that doesn't work actually in this case when you go back for a second the video i'm not going to do that but when you go back and plug both of these back in they actually both work okay so both of these are valid answers so when they ask us to find the measure of x y t we're going to plug it back in here if you plug the three in right here you're gonna get three times make sure you have to see this yeah you get three times three squared plus five times three plus seven you're going to end up with 49 degrees but also when you take the negative 15 force and you plug it in there you're going to get 3 times negative 15 fourths squared plus five times negative fifteen fourths plus seven you actually end up with 30.4 degrees and both of those answers guys are actually valid 49 degrees or 30.4 okay so that's kind of kind of a tricky question oh this video is one hour and five minutes okay so again you guys don't have to watch the whole entire video when i do these things they're they're they're just a resource for you i love you guys so much i want to make sure you guys have all the resources and information you guys need to be successful so hopefully you guys got something out of this and you're ready to go for the test and wish you guys the best of luck and we will see you on the next video