well welcome to the official start of calculus congratulations you made it in the class today we're going to talk about a review a review of a lot of the math 2 Concepts and some basic algebra Concepts that you really need to have down in order for you to be successful in calculus the first place we're going to start is section 0.1 chapter zero we're going to talk about lines just basic lines we're going to go through some families of Curves we'll get into some trig functions your favorite right and then we'll we'll continue to calculate so that's a weird lines there what's special about a line what do you know about lines two points well they have infinite points but specifically you need at least two right what else you know about lines they curve at all do they curve lines do the end what what do you need to know about a line in order to graph it need two points you need two points or you need one point and specifically one other thing SL you got to have SL lines have slope they're straight they don't end they have slope in fact the slope is what we're going to talk about for the first part of today the slope is pretty much how a line Rises or Falls now you might have been walked through this a long time ago we're going to go really quickly through how to find the slope of a line and we're going to invent the form formula ourselves so let's take a a generic line and we'll pick two random points on it we're not going to be specific on it cuz in order to create a formula you can't really say something specific like any particular point we need to work for all points can you tell me if I've got two points how can I I make a distinction between these what what does every Point have coordinates what are the coordinates somewhere over here what are what are the coordin generally for for a point x y okay so we know that any point that we draw is going to have the coordinates X Y this one will and this one will but the problem is we need some way to tell a difference between those those two points how are we going to do it what do you think num with some yeah with some numbers what are those what numbers one two are they going to go on top of our X or below our X place where now if I put them up here we're talking about x to the first power so we're going to put an x to the an X1 and an X2 a y1 and a Y2 saying that this is our our first point and our second Point whatever those points may be we can't use real numbers because then it wouldn't work in general for any two points can you tell me along the x-axis how far is is this one one X2 if if this is the point X1 y1 how far is that yeah it's X1 for sure how far is this point yeah are you guys with me on this no if you're you're all right so far if you're not go like this are we okay or no all right if this was like the point uh 3 comma 5 to get to 3 comma 5 You' go over three and up five right so then this would be three this is not 3 comma 5 it's X1 y1 so we're going over X1 how far are we going up good and here now when we talk about slope typically a long time ago when you were first introduced a slope you the teacher probably said yeah it's how your your your line Rises or Falls but then they also said slope is defined as what over what let's go ahead let's try to identify what our rise is and what our run is what would you say would be our rise this way this way so if we find the difference between those two numbers over there we're going to find the rise for our line what is the difference between those two numbers over there how do you find the difference this distance here sure yeah if this was 10 and this was three the distance between them would be what yeah You' subtract them right you do 10 minus 3 so here we're going to go well it's not 10 and three it's Y 2 and y1 so our rise we're going to call Y 2us y1 can we do the same thing with the Run how far is our run what's that distance represent or represented as21 I thought you were going to answer you instead you sneezed I was like she's and then darn it so for a good one yeah we got X2 - X1 for sure does that look familiar yeah if we use the letter instead of the word slope what letter we am I talking about here if we use M instead of slope we got our formula which is kind of you've seen it before right You' probably seen have you seen it invented before like this if you haven't well this is something new for you if you have well you've seen it again this is how you invent the slope formula the reason why we couldn't use specific points is because we wanted to be able to plug in any two points that I give you right so using that if you call your your points X1 y1 X2 Y2 you can find a slope for anything now the the one reason why I invented this for you is I want to show you that we can actually create an equation from that equation for a line from that slope formula so let's talk just a bit about equations what we're going to do is we're going to manipulate that formula by fixing one point we're going to be able to get the formula for a line here's what we're going to work with we're going to start with M = Y2 - y1 over X2 - X1 what I'm going to do is I'm going to fix this point say this is any specific one point and let the other one float what that does is it changes this formula into this formula instead of Y2 - y1 there's no more Y2 you see here's the thing a fixed point is a point with a little number under it a subscript like this would be a specific fixed Point X1 y1 this is a specific fixed Point X2 Y2 what I'm doing by fixing one point in only one point I'm saying that's the one point I'm fixing X1 y1 I'm going let the other one be floating so I'm going to Ras that uh Y2 and the X2 that means I'm going to have y - y1 over x - X1 that says one point's fixed this point is going to be formulaic it means you can plug in an X and get out of Y you ever seen an equation for a line not your head you've all seen it because you're all here right it has places for you to plug in X and get out y doesn't it so we need that inherently otherwise we don't have the equation for a line we have a specific two points we don't want that we want the equation for a line and here's a cool thing is there a way that you can solve for y - y1 how would you get rid of this denominator do what now someone said I just have bad ears see multip multiply it on one side or both sides so if I multiply this by oh that work cool x - X1 and over here x - X1 is this gone you guys are so quiet is this gone yeah they can't other for sure for sure on the right hand side tell me what I'm left with on the left hand side I'm going to reorganize this I'm not going to have x - X1 * m I'm going to have M * x - X1 you still okay with that so far multiplication is commutative doesn't matter what I have first what I have second no problem maybe you're not familiar with it written this way but I'll bet you've seen this before that one yeah what is that called no no it's not slope intercept point slope why is it called point slope mathematicians are very unoriginal it's called point slope because it's named after what you need to complete it you need a point and the slope that's why it's called point slope so by manipulating our slope formula interesting is it we used this we fixed only one point let the other one float worked around a little bit we now have point slope pretty useful stuff should we shall we do an example would you like to see something that we can use this stuff with I think you've seen it before this should be reviewed for you but let's go ahead let's see if we can first get the cobs out of your head because I know you weren't doing math over Christmas break were you or holiday break whatever you're doing were you doing math I was I was I was redoing this class to make it extra super special for you you should feel honored but let's go ahead and try to find the equation of the line that passes through these two points we'll get the cobs over our head we'll try to use a slope formula and then use point slope oh what what points so find the equation through these two points whenever your teacher taught you how to find the equation of a line they taught you you need you absolutely have to have two things you have to have one what you have to have one point very good and somebody else you also have to know the got to know the slope or be able to find the slope firstly do we have a point actually we have two of them we're set right do we have a slope is it given to you right now can you find it go ahead and find it by the way I usually do this I'll walk around the room if you need help at this point let me know because now would be a good time for me to help you what I want you to do right now is find the slope if you don't remember how to find the slope you assign X1 y1 to one point X2 Y2 to one point and plug it in that formula okay go ahead and try that right now for okay let's see what y'all did just want the slope for now work on the equation just a second so as far as the slope goes we need to pick one of these points to be X1 y1 another point to be X2 Y2 you just got to make sure it goes XY and it goes 1 one and 22 right those numbers got to be together and you have to have an alphabetical order you can't go YX which is going to be your X1 you all pick neg2 for your X1 doesn't matter it really doesn't matter does it I could pick this one as X1 but then that would have to be y1 so once you pick one letter the rest of them have to fall into place now typically people just want to make it easy and they put this one X1 y1 X2 and Y2 which I'm guessing most of you did that one the other way you you're going to get the same exact slope it's just your if you had a negative or a sign it's going to be in the opposite spot not a big deal let's go ahead and plug this in we have Y2 - y1 what am I going to write down if I'm supposed to do Y2 - y1 right now two like that okay good so we're subtracting but that also has a negative sign I'm be real careful about that I'm going to put in parenthesis to show that I'm subtracting a negative right there what is subtracting a negative what's that become great okay well yeah it ends up being addition and then for X2 - X1 we're going to have our 8 - -2 same idea what is our 2 - three how much we get over can you reduce slopes absolutely what is it okay quick show hands how many people were able to find the the slope good that's starting your everybody that's fine if you're not work on that later okay revisit this try to follow through this example see if you can do on your own and then come up that one half are we done we we're we're about halfway there okay we have the slope do we have a point still if we've got a point and we've got the slope we should be able to fill out point slope it's actually the same exact formula as slope we just have now fixed one point and we have the slope it's kind of good so we know we're going to be filling out Yus y1 check it out if you already have your y1 you can leave it if you want to make it easier on yourself and not deal with those negatives you could use that point that'd be fine as well it doesn't matter what point you have after you've already identified your slope here I'm going to stick with the same one to to keep it kind of continuous for us so we have y what am I gonna write perfect okay again I'm gonna use some parentheses saying that's a negative number in there equals I'm supposed to have my M what's my M uhhuh and X perfect let's clean this up just a little bit we're going to have y + 3 = 12 x + 2 if it's asking for point slope you know what you're done okay that that's it but this isn't exactly the easiest way to graph a line is it like that is there a way we could make it easier what would you do distribute sure we want to get rid of those parentheses we'll distribute that if we distribute the right side I'm still going to have this y + 3 but I'm going to get 12 x + what good 1 12 * 2 that's going to give us one and lastly less St is what yeah if I do that I'm going to get yal 12x that's something we're real familiar with we know how to graph that pretty well on XY AIS what is that called by the way yeah whenever we have y = mx plus b where we have some number time X plus or minus some constant we know that that's going to be that is slope intercept it's pretty easy to graph it gives you what you need to graph a line very quickly and again the reason why it's called SL intercept is that's what you have that M that's our slope what's the B stand for y intercept yeah good Y intercept can to graph it what we just how would you graph something with 12x - 2 can you tell me what is my Y intercept in this case so when we're graphing slope intercept that says if we have ative -2 that means we're going down or left which which what what's that mean down so we know our y says down two that says we're going to put a point right there that's where our minus 2 is coming from next we use our slope from the point that we just graph not the origin but the point we just plotted to find our next Point our slope is what was it is that up or down up up how many and over to the over to the right how many two yeah the positive or negative tells you whether you're going up or down you always go to the right so if we had negative - one2 that would say go down one but you're still going to the right you're always going to the right the plus plus negative tells you up or down you with me on this so far okay good so we're going to go up one y all said to the right two now that we have two points we know that two points delineates a specific line unique line we graph it we make sure we label this and we're done show hands how many people feel okay with graphing these lines using slope formula and the point slope good deal all right now there's a couple more things we got to talk about about lines before we talk about parallel and perpendicular which we're going to do in just a second if I told you that this was a line y equal to some some number c where C's constant what type of line is that let's just they're all straight lines right they're all straight lines that's a straight line well I think you might mean is is this a vertical line or is that a horizontal line horizontal line you know the way you can tell what variable do you have up there that means it's going to cross the y- AIS whatever variable you have says That's The Intercept that you have so there's no way to have this Line crossing the y axis does that make sense to you there's no way to do that the only way you can do that and have a constant line is like this that's a horizontal so if you have a y equals that's saying you're going to have a y intercept at that number and it's going to be horizontal so when y equals a constant you're talking about a horizontal line what's the way I can make a vertical line just straight up yep if x equals a constant we have an X intercept that's a vertical line another way you can think about that if you have y equals and you try to fit it into this formula the slope intercept of point slope you've got a slope of zero right there there's no slope there slope of zero is going to give you a horizontal line if you don't have a y though that means you're going to have an undefined slope that means you're going to be a vertical oh yeah let's manipulate one equation we'll see if we can put it into a slope intercept form we'll talk about some parallel perpendicular lines and then we'll go on to some trigonometry I can tell you're excited you excited yeah some of you are giving me death books right now you know that right here we have an equation that's kind of a standard form of a line it it'd be standard form if we added the three to one side and then we'd have we'd add the standard form is there a way you can put this into slope intercept go ahead and try that for me just go ahead and solve for y and make that slope intercept and want to make sure you can do it what would you do first if you're trying to solve that thing for y isolate yeah you're trying to isolate it what's your first step in doing that here add three you could add three then you could subtract 4X and then you could divide by two another way to do it you could subtract 2 y right it's probably going to eliminate a step for you if you subtract 2 y you're going to end with 4x - 3 = -2 y now typically we don't like to do that because we don't want to have a negative coefficient in front of Y but we just have to divide by a negative and if you're good with signs you can do that pretty easily how do we get that y by itself again just make sure if you divide you got to do it everywhere and you have have the same exact thing and you have to be good with your signs so 4X over -2 what are you going to get out of that how much 2 and then plus or minus how much equals sure maybe you write that a little bit different to get that in the form as you want it Y = -2x + 3 could you still graph that and as a matter of fact sometimes it's nice to keep it in stand I asked you to do it in slope intercept form but I also want to show you this if you have this in standard form which would be that have you ever learned the coverup method for graphing from standard form you ever seen that before if you want to find the x intercept cover up your yide by four you know this is going to cross the x axis at 4/3 it's easier than graphing a a fraction than going up and over down and over if you want to find the Y intercept you cover up the X divide by this number positive2 that's going to cross the y- axis at positive 3 halves you can graph a line like that as well so just a little refresher on those those show hands how many people feel okay so far on on our lines you still all right you you awake still this should be review and this is review for you I know some of you are thinking where's the calculus just wait for it all right the calculus is going to come whether you want to or not uh just hang on for a second enjoy the the nice slow stuff but really absorb this if you're if you're a little rusty on it what do you know about parallel lines that's actually the definition did you hear them over there we have the same exact slope that means we have parallel lines it's kind of like climbing stairs right the way stairs work is they're parallel that means you're going up and over the same rate otherwise on these stairs over here if your if your stairs didn't go up and over you're going to be like this you like oh These are nice these These are nice son of a gun you know if they were different slopes you they intersect somewhere we don't want that to happen for stairs we don't want that to happen for parallel lines so when we talk about parallel lines what we're talking about are lines that have the same exact slope do perpendicular lines have the same slope no no no they don't actually perpendicular lines meet at a very specific angle what angle do perpendicular lines meet at right so if one Line's like this the other line's got to be like that right means if one slope's positive the other slope's negative so we know that's going to come into play also there's a it's not just a negative slope that's going to be something like this right that's not going to cross at 90° we want to make it actually just a little kicked over how do I make it so it meets exactly 90° it's not only negative but also AAL very good a couple of you said reciprocal so perpendicular lines are lines which have are lines where the slopes are negative reciprocals of each other okay I was basic question if I give you an equation can you find a line that's parallel and or perpendicular sorry or perpendicular to a given equation can you guys do that let's try that out real quick that'll be our our last little timid review problem hopefully these have been timid for you we haven't even made it 30 minutes in we hav't we're going to get to some train in just a bit Yeah so let me say that I want to find the equation of the line that passes through this point and parallel to this find the equation passing through a set point and parallel to a given equation what's the two things you need for sure in order to make the equation of a line slope and a what now Point new point do I have a point cool do I have a slope do I have a slope right now no not yet I got to work on it but could you find that slope let's find the slope go ahead and do that on your own real quick give about 5 seconds you actually should be pretty good at this is the slope -2 no what am I missing that's the reason we talk about slope intercept right it's to find that slope very easily your slope's -23 now what I'm asking for is the equation of line that's parallel to that line but now goes through that point so what slope am I going to use if I want to find the the parallel line to this I'm use three halves I want the parallel to this oh parallel lines have the same slope so what we're going to write down we're going to find our slope we want the parallel slope so we're going to write down the parallel slope if it's parallel it's going to be exactly the same we know that our m is going to be -23 now what point do we use what point do we use speak up what point do we use okay does this for have anything to do with this problem actually all we cared about was finding the slope once we have the slope and we already identified a point wow we can just plug that into our point slope formula find the equation of our line so we'll do y - 7 = -23 x - 6 quick quick show hands how many people feel okay solving for slope give me a head now if you understand that the slope we are supposed to use is still -2/3 because we want to find a parallel do you see where the six and the Seven are coming from good deal can you work that out and make that slope intercept for me okay let's do that together I know that y - 7 that's going to stay there on the right hand side tell me what I'm going to get and then plus or minus what do you think plus how much good final step if we add set four because we got 2/3 * 6 2/3 * 6 you can simplify the fractions or get 12/3 that's four add our seven to both sides and we're done what if I change the problem and instead of having parallel I asked you to do perpendicular could you still do it let's talk about the the only changes that would occur up here okay I'm not going to change the whole problem I just wanted to walk you through it here if I'm talking about perpendicular would this process change would this change what's that going to become okay so this would become three so here now we're talking about perpendicular would the seven and the six change no we're just talking about slop about this is parallel perpendicular only has to do with the slopes this would change to our 3es this would change to our 3 x it'd be minus how much - 9 and if we added seven to both sides here we get 3 x - 2 so we can find both the parallel and the perpendicular slope do you feel okay about our basic line so far would you like to learn a little bit about angles of inclination and use a little bit of Tri trigonometry here and you're like no not really too bad we can do it anyway let's talk about angles of inclination we're going to use it at some point are there any questions before I race any of this stuff are you sure are we having fun yet wouldn't you rather be in here than out there in the rain some of you like no I'd rather be in the rain honestly right now prer that just lie to me just lie and say yes this is awesome Leonard I love this class you're the best thanks guys appreciate it if you did that I would just watch this video over and over again and have you go thanks Leonard you're the best yeah thanks Leonard you're the best yeah thanks lard you're the best see I just have I do it with my workout thanks lard you're the best bam B bam so cool I know I'm the door oh my angle of inclination let's talk about how we can use angle of inclination and relate it to a slope we're going to start with some line random by what I talk about the angle of inclination what we talk about is the angle that any line makes with the x-axis so the angle of inclination the these actually would be the same angle if you taken geometry you know that those those are the same we're really just talking about this one though we call it f we want to find some way to represent this line as having that angle if we think about the X and the y axis notice noce that this we could represent as a change in X you guys have seen that that terminology before the Delta X Change in X and this would be well the change in y now think back to your trig days it's just basic trigonometry is there a trig function that that uh that relates this angle and these two specific sides remember this would be a 90° angle what does that what what's if this is a triangle which it is uh what's this side called that's hypotenuse what's this one according to this angle that's the good and this one is the which one relates adjacent and opposite tent not sign sign would be opposite over hypotenuse tangent does or coent we don't deal with cotangent Cent's adjacent over opposite we want to deal with probably the easy one tangent if we talk about tangent the tangent of that angle is equal to the opposite over the adjacent not your head if you're okay with the the tan opposite over adjacent if you're not you you're definitely going to want to review your trigonometry before attempting this class you're deal with a lot of trigonometry but here's the cool thing what what is what do we always already Define as as change in y over change in X or rise over run what do we always already Define that as so then we we have this relationship we know that tan Theta well that's Delta y Delta X but this is also the same thing as slope or M if you bring all this together slopes equal to 10 do you see the relationship between your slope and your angle of inclination it's same well it's the same as the tangent of that angle of inclination because Tan's defin is opposite over Jason and so is slope we can make that job what's kind of cool is it says that if you know the angle well you can find the slope can't you if you know the angle you can do that in a calculator you'll be able to find the slope if you know the slope you can find the angle those things are intertwined they're they're equal to each other along with that tangent so let's shall we try one would you like to see an example of how this is done would you you're the most mellow class I've ever had would you like to see one bring it up let's say that your angle is 30° someone quickly 30° as radians is what same thing as 5 six yeah I want you to find the slope of the line that has an angle of inclination of 30° or/ 6 here's how you do it we know for a fact that m equals tan Theta don't forget that okay that that's your equation that's what you do now you know that the slope is equal to the tangent of that angle what's our angle what's our angle 30 30° so if I plug in my 30° or I plug in my Pi 6 at the same angle okay if I plug that in there I know the slope I'm looking for is equal to the tangent of Pi 6 or 30° whatever you want to work in I like Pi 6 now all those with your your uh unit circle tattooed on your right arm okay look down there can you tell me what tangent of pi/ 6 is can you tell me what tangent remember tangent you have to define a sign over cosine right so you got to find s of 30° or S of Pi / 6 put it over cosine or have a memorized put it over cosine of Pi / 6 or 30 degrees look at you in a circle if you have one you should have one need one honestly you should tattooed on your forehead backwards that way you look in the mirror you can memorize it it's a good idea good I haven't done it personally but I'm waiting for someone to actually do that pretty classic tangent of pi over 6 is s over cosine the S of Pi / 6 um 12 cosine of pi 6 < tk3 over two that's what those correct me if I'm wrong but I think those are right yeah pretty sure I'm doing this on the spot good all right can you simplify that a little bit yeah those twos are actually going to cross out uh you you you know divide those fractions complex fractions you're going to flip that multiply the twos are going to be gone what you're going to end up with is 1 over the < TK 3 or if you rationalize the denominator you know how to rationalize denominators right multip byun3 ro3 you're going to get 3 < tk3 3 strangely enough that's your slope right there your slope is < tk3 all right uh let me recap did you get that that was funny let me recap oh come on stand up next time maybe you'll get it uh here's I I honestly will recap though here's what you do to find the slope if you have the angle of inclination you take your angle you plug it in and you figure out the tangent of that angle that's honestly it I mean if you can find the tangent of pi/ 6 you have your slope that that is your slope all right now can you go backwards that's going to be another question for us what if I said I have now room I have a slope of1 can I find the angle of inclination we we only have one equation the only thing that we know is that the slope equals tan Theta over here we knew the Theta right we were looking for the the the slope the M over here we know which one we know the m so notice we're using the same exact equation here we knew our angle you could have put 30° here very easily and done the same exact thing here we know our M we know1 equal tan thet how do we find Theta you could do tan inverse sure of both sides take it to the left we do tan inverse of1 equals if you do tan inverse of both sides uh it would look like this You' have tan inverse you'd have tan inverse tan inverse of tangent gone you have Theta tan inverse of1 that's what we have right here this is what the question asks you okay here's here's this in plain English it says I want you to tell me the angle so that when I take tangent of it it's going to give give me NE 1 that's what tan inverse says it's it's kind of a backwards way of of looking at it it's saying find me the angle that way when I take the tangent of it it's going to give me the value of negative 1 do you understand the question there it basically it goes down to look at the unit circle find out where s and cosine are the same but have different signs because you know tangent is s over cosine right so you look at your unit circle find out where it's um oh what we have it's two of them I don't remember the exact one but find find out where you have S and cosine exactly the same off by sign that will give you Nea 1 have you found do you have your unit circle out yeah 3 Pi that's exactly right oh yeah have right here so this happens where your s your cosine are the same but off by a sign that's uh < tk2 over 2 over < tk2 over2 that's where that happens on your unit circle so check that out later if you don't have your unit circle handy you're going to have that same exact value only this one's going to be negative that happens twice actually and if you're confused is well well wait a second don't I get two of these same values yeah you do but think about what you're actually doing you're trying to find the slope of a line that goes like that right it's going to cross two quadrants it's going to have the slope here and there one's going to be positive one's going to be a negative uh way of looking at an angle they're all the same though uh 3 Pi what you say 3 Pi 4 3 pi over 4 or or the negative version of that uh used as a reference angle um so here we go we got that negative 1 so we know that this happened when Theta was equal to 3i 4 or if you want to translate that that's uh 135° so either way we look at this we can find slopes from angles can find angles from slopes I'd say that this one's probably a little bit more trick some for you cuz you're going to actually have to do a little bit of work on it this you could probably just plug in a calculator if you memorize like the 0877 thing you know oh that's like three over two yay or whatever you did zoom up at me I memorize those I don't want to look them up uh but if you if you do this on a calculator it'll actually work out for you it probably won't give you the square root of three but you can figure that out here you're going to have to use a unit circle if they give you a slope you're going to have to find the angle to which s over cosine makes that value it's going to be some you might have two of them it's not going to matter that the same exact value of the angle choose either one of them it'll it'll work out the same but find the angle to which you're getting that value do you guys see that the process here I know I went very quickly through this do you see the process yeah or no how feel okay with with this so it take a little bit of work to get handy on that yeah probably probably a little bit more work to out um let's see one more thing I want to go over let's talk about the distance formula real quick it'll wrap up our very first section here are you sure there's no questions on angles of inclination anymore we got time so if I give you this on a test and I say I want you to find the slope if the angle is pi over 3 you do it could you do it if you had a unit circle oh okay that's question uh if I say the slope is 1/2 can you find tangent where the angle would give You2 could you do it with the unit circle okay practice that stuff that's what I'm looking for I a test we get I don't care sure to be honest with you this is very much review Stu uh what I'm trying to get you to get back familiar with the tangent s cosine secant cosecant cotangent ideas because we're going to move way past this we're going to be using this within some problems all right so this isn't going to be a problem this is going to be within some problems you get it it's kind of like factoring isn't everything in algebra you just use factoring in everything in algebra you you get the analogy so that that's kind of what what we're doing here we're going to use a lot of trigonometry in this class we're going be doing things called derivatives and integrals and all going to involve trig functions so you got to know the trig pretty well to be successful uh people say that you go to calculus to finally fail algebra and trigonometry you made it this far but it's not the calculus going to hold you back I promise it's going to be your algebra and it's going to be your trigonometry I guarantee it the calculus is actually quite easy uh it's those Concepts put together with Calculus and makes kind of hard so if you're good at algebra in trig you're can be fine absolutely fine you stick with this class okay shall we distance formula in about a minute and a half then we'll call it a day let's do distance formula we're going to do it the same way that we did our slope formula which is we're going to pick two random points X1 y1 and X2 Y2 only this time we're going to find the distance distance between them if we have the X1 y1 and X2 Y2 well we for sure know that's X1 and that's X2 and this is y1 and that's oh that's y 2 what we want to do now though is use something that relates this side this side and that side and that's a right triangle what relates to thatan Pythagorean theorem absolutely you're right if we call this our distance here's what we can say we know that this length is X2 - X1 by the same step stuff that we just did with slope formula we know that this distance is Y2 - y1 by the same stuff we just used on our slope formula this is the length this is the length from those corresponding points we want to find the distance if we do Pythagorean theorem we've got D2 equals what what's pagan theorem say sure yeah a s + b s = c s some of you know that or I prefer a leg squ plus a leg squ equal a hypoten Square because that that tells you what you're doing right so if we take a leg squared that's this and a leg squared that equals the hypoten squar I already have that here's my first leg that's the distance squared plus the second leg that's the distance squared can you guys see the Pythagorean theorem at work here you guys see the leg Square yeah this is this is one leg right I'm just squaring it here's another leg I'm just squaring it and that has to be by Pythagorean theorem equal to the hypotenuse squared so we got that the only thing we need to do now get rid of the square how do I get rid of the square yeah that just means the whole thing now we are going to omit the plus minus because well we can't have a negative distance that doesn't make any sense so we're just going to have the square root of this entire thing D equals by the way oh here's a good question for you see where you're at will this square root get rid of this square and that square no what do you think does it does it work that way across [Music] addition this multiplication sure addition no way no.0 can't do that and that's our distance formula you know what I'm not going to do an example for because it works really really really similarly to our slope formula would you be able to if I gave you two points would you be able to find me an X1 and a y1 and an X2 and a Y2 and plug them in and not mess the science up right you just Square this value you square that value you add them of course it's going to be positive right because you're squaring something squaring something and adding it just don't forget to take a square root uh you can either leave it in terms of a square root or approximated give me a decimal answer how many people feel pretty good about what we talked about today all right that's good m