Calculus 1 Lecture 0.1: Lines, Angle of Inclination, and the Distance Formula

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 two 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 zero point one chapter zero we're going to talk about lines just basic lines we're going to go through some families of curves we'll give you some trig functions your favorite right and then we'll we'll get to the capital itself lots of weird minds what's special about a line what do you know about lines love it have infinite points but specifically you need at least two right why don't you give up lines they curve it all do they curve lines do the end what what do you need to know about a line in order to graph it you need two points or I need one point and specifically one of the lines have slope they're straight they don't end that 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 formula ourselves so let's take a generic line and we'll pick two random points on it we're not going to be specific on it because it 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 make a distinction between these well what does every point have coordinates what are the corners somewhere over here what are the corners generally for a point excellent 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 with some VI with some numbers what are those what numbers are they going to go on top of our X or below our X and please we're done if I put up here we're talking about X to the first power so we're going to put an X to the X 1 and X 2 a 1 1 or Y 2 saying that this is our our first point in our second point whatever those points may be we can't use real numbers because then it wouldn't work in general 42 points can you tell me along the x-axis how far is is this one 1 if this is the point x1 y1 how far is that yeah it's x1 for sure how far is this point yeah okay is within this budget if you're all right so far if you're not Gilligan some we okay or no all right if this was like the point of 3 comma 5 to get to 3 comma 5 you go over 3 and up five right so then this would be three this is not 3 12 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 smoke typically a long time ago when you're first introduced the slope the teacher probably said yeah it's how your line Rises or Falls but then they also said slope is defined as 1 over what let's go ahead let's try to identify what our Rises 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 right sure yeah if this was 10 and this was 3 the distance between them would be what okay you'd subtract it right you do 10 minus 3 so here we're going to go well it's not 10 and 3 its y2 and y1 so our rise we're going to call y2 minus 1 1 can we do the same thing with the run how far is our run what's that distance represent the representatives taking you - one I thought you're going to answer you instead you sneezed I was like she's off and then John it it's a pretty good one yeah we got x2 - x1 for sure sound familiar yeah if we use the letter instead of the word slow what letter - am i talking later we use em and step slope we got our formula we're just kind of you've seen it before right you probably saw anything invented before like this if you haven't well something new for 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 want to be able to plug in any two points that I give you right so using that if you call your points x1 y1 x2 y2 you can find a slope for anything now the one reason why I invent 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 lines so here's what we're going to work with we're going to start with M equals y2 minus y1 over x2 minus 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 minus y1 there's no more y2 you see here's the thing a fixed point is a point with a little number under 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 only one point I'm saying that's the one point I'm fixing x1 y1 I'm let the other one be floating so I'm going to erase that y2 and x2 that means I'm going to have Y minus y1 over X minus x1 that says 1 points fixed this point is going to be formulaic it means you can plug in an X and get out of Y you ever seen the equation for a line not your head you've always seen it because you're all here right it has places for you to plug in X and get outlined in 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 equation for a line here's the cool thing is there a way that you can solve for y minus y1 how would you get rid of this denominator do what someone said I can just will have bad ears multiply on one side or both sides so if I multiply this by all don't work cool X minus x1 and over here X minus x1 is this gone you guys are so quiet is this gone and I can't ever know 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 minus X 1 times M I'm going to M times X minus x1 you still okay with that so far multiplication is commutative didn't matter what I have first we have second Oh problem maybe you're not familiar with it written this way but I'll bet you've seen this before work yeah what is that called just think 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 slope this what's called point slope so by manipulating our slope form is interesting in that we use this we fixed only one point let that of a float worked around a little bit we now have point slope pretty useful stuff shall we shall we do an example would you like to see something that we can use the stuff would I think you've said before it should be reviewed for you but let's go ahead let's see if we can first get the cobwebs out of your head because I know you weren't doing math over Christmas break we're or holiday break whatever you're doing what are you doing math I was I was I was redoing this class to make an extra super special for you you should feel honored but let's go ahead and try to find the equation of the line passes through these two points look at the couples or a head will try to use a slope formula and then use point slope with 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 absolutely have to have two things you have to have one what if F one point very good and something else you also have to know the get little slope or be able to find the slope firstly do we have a point actually a two of them we're set right do we have a slope is it given to you right now can you find it you know go ahead and find it by the way you should do this I'll walk around the room if you need help at this point let me know is now would be a good time for me to help you what it wants 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 go ahead and try that right now Oh okay let's see what y'all did I just want the slope for now we 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 just got to make sure it goes X Y and it goes 1 1 & 2 2 right those numbers got to be together and you have to have an alphabetic order you can't go Y X which is going to be your x1 do pick negative 2 for your x1 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're going to get the same exact slope just sure if you had a negative or signs can be in the offensive spot not going to you let's go ahead and plug this end we have y2 minus y1 what am I going to write down if I'm supposed to do y2 minus y1 right now - like that - okay good so we're subtracting but that also has a negative sign I'll be real careful about them and put it in parentheses to show that I'm subtracting an 8 what is subtracting negative what's that become great okay flood yeah it ends up being addition and then for x2 minus x1 we're going to have our 8 minus negative 2 same idea what is our 2 minus negative 3 times we get over can you reduce slopes absolutely what is it a quick show hands how many people were able to find the slope good start here but that's fine if you're not work on that later okay revisit this try to follow through this example see if you're doing it own and then come up that 1/2 I'll be done we're about halfway there can we have the slope do we have a point still if we've got a point that we get 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 it's kind of good so we know where you thought Y minus y1 check 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 can use that point that define 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 keep it kind of continuous for us so we have Y what am I gonna write perfect okay again I will use some parentheses saying that's a negative number in there equals I'm so sad my M what's my M and X perfect let's clean this up just a little bit we're going to have y plus 3 equals 1/2 X plus 2 if it's asking for point-slope you know what you're done it 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 can make it easier what would you do sure we want to get rid of those parentheses we'll distribute that if we just hit the right side I'm swimming am is y plus 3 but I'm going to get 1/2 X plus 1 good 1/2 times 2 that's going to give us 1 and lastly less tip is what yeah if I do that I'm going to get y equals 1/2 X that's something we're real familiar with we know how to graph that pretty well in X Y axis what is that call by the way yeah whenever we have y equals MX plus B when we have some number times X plus or minus some constant we know that that's going to be called with 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 slope-intercept is that's what you have that animal well that's our slope what's the B stand for yeah I could find yourself can you graph it let me just graph how would you grab something with 1/2 X minus 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 a negative 2 that means we're going down or left which what that means so we know our wise is down to and 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 with the point we've just plotted to find our next point our slope is what was it is that up or down up how many and over to the over to the right how many 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 1/2 that would say go down 1 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 have known this so forth ok so we're going to go up 1 you offset to the right 2 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 we're done show hands how many will feel okay with graphing these lines using slope formula and the point slope good day all right now there's couple more things we got to talk about about lines before we talk about parallel and perpendicular we're going to do it just second if I told you that this was a line y equals to some some number C where C wants 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 but I think you might mean is is this a vertical line or is that a horizontal bottom or song line you know the way you can tell what variable do you have up there that means it's going to cross the y-axis 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 though the 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 once the way it can make a vertical line just straight up yep if x equals our constant we have an x intercept that's a vertical line we can think about that if you have y equals if you try to fit it into this formula the slope intercept the point slope you've got a slope of 0 right there there's no slope there slope is 0 is going to give you a horizontal line if you don't have a Y though that means you're going to have a undefined slope I mean you're going to be a vertical oh yeah let's manipulate one equation we'll see if we can put it into slope-intercept form we'll talk about some parallel perpendicular lines and then we'll go onto some trigonometry I can tell you're excited excited yeah some of your 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'd be standard form if we added three to one side and then we'd have we'd have 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 make up slope-intercept we want to make sure you can do it what would you do first if you're trying to solve that thing for wine now you're trying to isolate it what's your first step in doing that here you could add three then you could subtract 4x and then you can divide by two another way to do it you could subtract two Y right it's probably going to limit it a step where you subtract two Y you're going to end with 4x minus 3 equals negative 2y now typically we don't like to do that because we don't want to have a negative coefficient for the Y but we just have to divide by negative if your goodness 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 to have the same exact thing and you have to be good for signs so 4x over negative 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 skipping in the homes you want it y equals negative 2x plus 3 halves could you still graph that as a matter of fact sometimes it's nice to keep it in say I asked you to do it in slope-intercept form I want to show you this if you have this in standard form which would be that have you ever learned the cover-up method for graphing from standard form you ever seen that before if you want to find the x-intercept cover up your Y divide by 4 you know it's going to cross the x axis at 4/3 it's easier to graph a fraction and going up over down over if you want to find the y-intercept you cover up the X divided by this number two positive two that's going to cross the y axis at positive three-halves you can graph a line like that as well so just a little refresher on those Johannah some people do okay so far on on our lines still all right you'll be awake still this should be review 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 just hang on for a second enjoy the nice slow stuff but really absorb this in here if you're a little rusty on it what do you know about parallel lines that's extra 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 these stairs over here if you're if your stairs didn't go up and over you give me like this you're like oh these are nice you feed to my 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 we talk about parallel lines when we're talking about our lines that have the same exact slope you do perpendicular lines have the same slope no no no actually perpendicular lines meet at a very specific angle what angle do perpendicular lines meet at right so if one lines like this the other lines got to be like that right means if one slopes positive those slopes 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 like an across a 90 degrees we want to make it actually just a little kicked over how do I make it so it meets exactly the 90 degrees is not only negative but also a very good W so reciprocal so perpendicular lines are lines or chats or lines where the slopes are negative reciprocals of each other okay let's basic question if I give you an equation can you find a line that's parallel and/or perpendicular I'm sorry or perpendicular to a given equation can you guys do that let's try that on real quick that will be our last little timid review problem hopefully these have been timid for you and they've made it 30 minutes and we haven't we're going to get to some training just a bit you get up 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 it ought to make the equation of a line slope and what no dew point do I have a point cool do have a slope do have a slope right now not yet I got to work on it but could you find that slope let's find the slope go ahead and do it on your own real quick in about five seconds you should be pretty good at this is the slope negative to know what am I missing as a reason we talk about slope-intercept right and to find that slope very easily your slopes negative two thirds 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's slope my going to use if I want to find the the parallel line of this amuse three halves but won't the parallel is this all parallel lines have the same slope so what we're going right down we're going to find our slope we want the pair little 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 negative 2/3 now what point do we use what point do we use speak up for these does this Ford have anything to do with this problem actually all we cared about was finding stuff once we have the slope and we've already identified a point wow we can just plug that into our point-slope formula find the quays of our line so we'll do y minus 7 equals negative 2/3 X minus 6 quick quick show of hands how many will feel ok solving for slope give me a head not if you understand that the slope we're supposed to use is still negative 2/3 because we want to find a parallel do you see with a 6 and 7 or coming from good deal can you work that out and make that slope intercept for me ok let's do that together I know that Y minus 7 that's can stay there on the right-hand side tell me what I'm gonna get and then plus or minus what you think plus how much good final step if we add set 4 because we have 2/3 times 6 2/3 times 60 and some of the fractions or your 12 thirds that's 4 add our 7 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 only changes that would occur up here okay I'm not gonna change the whole progress I want to walk you through it here I'm talking about perpendicular with this process change would this change what's that going to become okay so this is becoming three now so here now we're talking about perpendicular with the seven and the six change now we're just talking about slope about this is parallel perpendicular only has to do with slopes this would change to our three halves this would change to our three halves X and B - how much Y it's nine if we added seven to both sides here we get three halves X minus two so we find both the parallel and the perpendicular slope do you feel okay about our basic lines so far would you like to learn a little bit about angles of inclination and use a little bit of trigonometry here and you're like no not really do that you do anyway let's talk about angles inclination we're going to use it at some point are there any questions before erase any of this stuff are you sure are we having fun yet once you rather be in here that out there in the rain some of you're like no I'd rather be in the rain right now just lighten II just lied say yes this is awesome Leonard I love this class you're the best thanks guys beginning 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 Leonard you're the best yeah yeah I do look with my workout thanks Larry you're the best damn damn so cool all right Oliver Oh my angle of inclination let's talk about how we can use angle of inclination and related to a slope we're going to start with some line random line when 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 these actually be the same angle taking geometry you know those ones are the same we're really just talking about this one call it theta 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 that this we could represent as a change in X you guys have seen that that terminology for the Delta X change in X and this would be well the change in Y now think back to your trig days is just basic trigonometry is there a trig function that read that that relates this angle and these two specific sides remember this would be a 90-degree what does that what's it this is a triangle which it is what's this cycle 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 engine not science I would be opposite over hypotenuse and tangent does or cotangent we don't do a cotangent cotangent 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 now you're going to be okay with the tan opposite over adjacent if you're not you you're definitely going to want to review your trigonometry for attempting this class or do with a lot of trigonometry but here's the cool thing what what is what do we always already define as change in Y over change in X or rise over run what I've always already defined that as so then we have this relationship we know that tan theta that well that's Delta Y over Delta X but this is also the same thing as slope or M if you bring all this together slope is equal to tan theta do you see the relationship between your slope and your angle of inclination it's same for it well it's the same as the tangent of that angle of inclination because tans define opposite over adjacent and so is slope we can make that jump what's kind of cool is it says that if you know the angle well you can find the slope get you if you know the angle you get a calculator you build find slope if you know the slope but you can find the angle those things are intertwined they're they're equal to each other along with that tangent so let's so we try one which likes to example how this is done would you yeah here are the most mellow class I've ever had which like c1 let's say that your angle is 30 degrees someone quickly 30 degrees as radians is what good that's same things five or six I want you to find the slope of the line that has an angle of inclination of 30 degrees or PI over six here's how you do it we know for a fact that M equals tan theta don't forget that okay 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 degrees so if I plug in my 30 degrees or I plug in my PI over 6 at the same angle okay I plug that in there but I know the slope I'm looking for is equal to the tangent of PI over 6 or 30 degrees wherever you want to work in I like PI over 6 now all those with your your unit circle tattooed on your right arm okay look down there can you tell me what tangent of PI over 6 is and you tell you intangible remember tangent you have to define a sine over cosine right so you can find sine of 30 degrees or sine of PI over 6 put it over cosine or may have a memorized put it over cosine of PI over 6 or 30 degrees look at your circle you have 1 you should have a neat left honestly you should tattooed on your forehead backwards that way you look in the mirror you can memorize it it's a good idea it's good I haven't done it personally but I'm waiting for some of the apps to do that pretty classic tangent of PI over 6 is sine over cosine the sine of PI over 6 1/2 cosine of PI over 6 root 3 over 2 that's because it should correct me if I'm wrong but I think those are right or should I'm doing this well this one good all right can you simplify that a little bit yeah those twos are actually going to cross out you you know divide those patches complex fractions you're going to flip that multiply the twos are going to be gone which can end up with is 1 over the square root of 3 or if you rationalize the denominator you know that there's right multiply the root 3 root 3 to get 3 root 3 root 3 strangely enough that's your slope right there your slope is root 3 all right let me recap did you get that that was funny let me recap come on stand up next time if you'll get it here so 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 angle that's honestly it I mean if you can find the tangent of PI over 6 you have your slope that that is your slope all right now can you go backwards let's get another question for us what if I said I have now I have a slope of negative 1 can I find the angle of inclination 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 slope then over here we know which one we know the M so no series of same exact equation here we knew our angle you could have put 30 degrees here very easily than the same exact thing here we know our M we know negative 1 equals tan theta how do we find theta you can do tan inverse shirt on both sides take it to the left we do tan inverse of negative 1 equals theta if you do tan inverse of both sides you have ten inverse you get a ten inverse tan inverse of tangent gone you have failed tan inverse of negative one that's what we have right here this is what the question asks you okay here's there'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 me negative one that's what tan inverse says it's it's kind of a backwards way of looking at it and saying find me the angle that way when I take the tangent of it it's going to give me the value of negative ones you understand the question there it basically goes down to look at the unit circle find out where sine and cosine are the same but have different signs because you know tangent is sine over cosine right so you look at your unit circle find out where it's or would we have it's two of them I don't know this anyone but find and find out where you have sine and cosine exactly the same multiply sine that will give you negative one have you found to have your unit circle oh yeah that's exactly right oh yeah that's right here so this happens where your sign your cosine are the same but off by a sign that's root 2 over 2 over root 2 over 2 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 if your confuses we'll 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 a slope of a line that goes like that right it's going across two quadrants it's going to have the slope here and there one's going to be positive one's gonna be a negative way of looking at all the same though 3pi which is say 3 PI over 4 3 PI over 4 or the negative version of that use as a reference angle so here we go we got that native one so we know that this happened when theta was equal to before or if you want to translate that that's a 135 degrees so either way we look at this we can find slopes from angles we can find angles from slopes I'd say that this one's probably a little bit more tricks them for you because you're going to actually have to do a little to work on this you can probably just plug in a calculator if you memorize like the point eight seven seven thing you know that's like three or two VA but whatever you did you zoom up to me and I memorize those I don't want to look them up but if you do this on a calculation workout for you 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 sine over cosine makes that value it's going to be something you might have two of them it's not going to matter you that the same exact value of the angle choose either one of them to work out the same but find the angle to which you're getting that value do you guys see that the process here knowing what very quickly through this do you see the process yeah I know how we will go okay with with this so you take a little bit of work to get handy on that 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 are you sure there's no questions on things of inclination anymore get time so if I give you this on test and I say I want you find a slope if the angle is PI over three you do it could you do it get a unit circle okay if I say the slope is one-half can you find tangent where the angle would give you one half could you do with your circle okay practice that stuff that's not in test I don't care sure to be honest with you this is very much reduced up what I'm trying to get you to get back familiar with the tangent sine cosine secant cosecant cotangent ideas because we're going to move way past this where to be using this within some problems all right so this isn't going to be a problem it's going to be within some problems you get it's kind of like factory isn't everything in algebra you just use factoring in everything in algebra you give the analogy so that that's kind of what we're doing here we're going to use a lot of trigonometry in this class we're be doing things called derivatives and integrals and they're all going to involve trig functions so you've got to know you trade pretty well to be successful people say that you go to calculus to finally fail algebra and trigonometry have you made this bar 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 okay I guarantee at the calculus is actually quite easy it's those concepts put together with calculus in X comma R so if you're good at algebra trig you can find absolutely fine you stick with this class ok shall we distance formula in about a minute and a half and with a little bit let's do this as 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 if we have the x1y1 x2y2 well we for sure no that's X 1 and that's X 2 and this is y 1 and that's so that's why to 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 that fagor in theorem absolutely you're right if we call this our distance here's what we can say we know that this length is x2 minus x1 by the same stuff that we just did with slope formula we know that this distance is y2 minus y1 by the same stuff we just use our slope formula this is the length this is a length from those corresponding points we want to find the distance if we do Pythagorean theorem we've got D squared equals what's by Thakker daresay sure yeah a squared plus B squared C squared some you know that or I prefer a leg squared plus a leg squared equals the hypotenuse squared is that that tells you what you're doing right so if we take a leg squared what's this add a leg squared that equals the hypotenuse squared 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 do you guys see the leg squared 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 this 1 and that just means the whole thing now we are going to omit the plus minus because well we can't a negative distance as a making sense so we should have the square root this entire thing D equals by the way oh here's a good question for you see where you're at well this square root get rid of this square in that square what do you think does it work that way across addition this multiplication shirt addition new way no way no can't do that and that's our distance formula and you know what I'm not going to do an example for you 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 y1 and an x2 and y2 and plug them in and now Vanessa signs up right you just square this value you square that value add them of course it's going to be positive right because you're squaring something squaring something to adding it just don't forget to take a square root you can either leave it in terms of square root or approximate it give me a decimal answer how many will feel pretty good what we talked about today all right that's good

Channel: Professor Leonard

Views: 1,357,072

Rating: 4.9573393 out of 5

Keywords: Leonard, Lecture (Type Of Public Presentation), Calculus (Concepts/Theories), Trigonometry (Field Of Study), Angle (Dimension), Mathematics (Field Of Study), Line (Literature Subject)

Id: fYyARMqiaag

Length: 48min 58sec (2938 seconds)

Published: Tue Jan 24 2012