01 - The Distance Formula, Pythagorean Theorem & Midpoint Formula - Part 1 (Calculate Distance) - By Math and Science
Transcript
| 00:00 | Hello . Welcome back to algebra . The title of | |
| 00:02 | this lesson is the distance formula , the midpoint formula | |
| 00:06 | and the pythagorean theorem . This is part one . | |
| 00:08 | We have several parts to these lessons . So , | |
| 00:10 | as you can see from the title , we're going | |
| 00:12 | to cover a lot of material in one lesson . | |
| 00:14 | Now , most of you , probably everyone watching this | |
| 00:16 | lesson has had some exposure to the Pythagorean theorem before | |
| 00:19 | . You've also probably had some exposure to the distance | |
| 00:21 | formula before , and some may or may not have | |
| 00:24 | had some exposure to the midpoint formula . What we're | |
| 00:26 | gonna do is first we're gonna review and talk about | |
| 00:28 | what that pythagorean theorem is , why it's important . | |
| 00:30 | And then we're going to show you that the distance | |
| 00:32 | formula that we use in algebra and we're going to | |
| 00:35 | learn in this lesson . It's basically a direct extension | |
| 00:38 | , it comes from the Pythagorean theorem , so it's | |
| 00:40 | almost like the same exact thing . And I'm gonna | |
| 00:42 | show you that a lot of times students don't understand | |
| 00:45 | that the distance formula is just nothing more than what | |
| 00:47 | they already understand in the Pythagorean theorem , we'll also | |
| 00:50 | talk about this midpoint formula , so we're getting kind | |
| 00:52 | of into coordinate algebra , coordinate geometry , there's a | |
| 00:55 | little bit of overlap between what we're learning now and | |
| 00:57 | what we've learned in geometry in the past , but | |
| 01:00 | we're gonna go a level deeper because we're into the | |
| 01:02 | more kind of advanced algebra here , we're gonna go | |
| 01:04 | a little bit deeper . I'm going to also take | |
| 01:05 | the opportunity to show you why this stuff is so | |
| 01:08 | important to modern science and math . I want you | |
| 01:11 | to understand that the things that you're learning are not | |
| 01:13 | just useless things , they're extremely important even to modern | |
| 01:18 | science , Modern physics , modern Chemistry , modern Engineering | |
| 01:21 | . And so I'm gonna get into it as we | |
| 01:23 | get into the lesson a little bit more , But | |
| 01:25 | just as a kind of an advanced preview , right | |
| 01:27 | , this concept of the distance formula that we're going | |
| 01:30 | to learn here , um it really only as we're | |
| 01:32 | going to learn in this lesson , it applies to | |
| 01:34 | when we draw pictures on a flat sheet of paper | |
| 01:36 | , or we draw pictures on a flat board like | |
| 01:39 | this , we can calculate the distance between any two | |
| 01:41 | points . That's what the distance formula does , right | |
| 01:44 | ? If you've studied it in the past , you | |
| 01:45 | know that that's what it does . In other words | |
| 01:47 | , I can put two points on the board and | |
| 01:48 | I can figure out how many centimeters are between those | |
| 01:50 | two points If I set up a coordinate grid and | |
| 01:53 | go from there . Now in 1915 , someone you've | |
| 01:56 | probably heard of , Albert Einstein proved what we call | |
| 01:59 | now the general theory of relativity . It's Einstein's theory | |
| 02:03 | of gravity , right ? So you might think , | |
| 02:04 | why are we talking about gravity and an algebra lesson | |
| 02:06 | ? It's because when you really dig into the details | |
| 02:09 | of relativity theory , which is one of the crowning | |
| 02:12 | achievements of modern physics , right ? That the understanding | |
| 02:15 | of of gravity not being a force between things , | |
| 02:19 | but gravity being the curvature of space and time . | |
| 02:22 | You probably heard that curvature of space and time curvature | |
| 02:24 | of what we call space time . When you drill | |
| 02:27 | down into that theory into the nitty gritty details and | |
| 02:29 | advanced physics , what you're going to find out is | |
| 02:32 | the way that you measure distances in space . Time | |
| 02:35 | is very similar to this distance formula that you're going | |
| 02:38 | to be reviewing in this lesson right now . So | |
| 02:41 | we're gonna be covering how to calculate distances between things | |
| 02:44 | but just keep in the back of your mind something | |
| 02:45 | that you think is simple like this is something that | |
| 02:48 | Einstein worked on for many , many years to prove | |
| 02:50 | that when space and time or curve , which you | |
| 02:53 | can measure by using this distance formula , that gives | |
| 02:55 | rise to what we actually call gravity here . And | |
| 02:59 | I know that that's beyond the scope of an algebra | |
| 03:00 | class , but that's something I want to point out | |
| 03:02 | because it shows you that the things that you're learning | |
| 03:05 | now have real uses for real science and advanced physics | |
| 03:10 | and chemistry and engineering . So let's dive into it | |
| 03:13 | . We're going to recall something before we get into | |
| 03:15 | the distance formula , we're going to talk about and | |
| 03:17 | review something we call the pythagorean theorem , pythagorean serum | |
| 03:24 | . And I'm going to spend obviously time talking about | |
| 03:28 | the Pythagorean theorem and the distance formula , all that | |
| 03:30 | . And then I'm going to show you a little | |
| 03:31 | bit more about this curving of space and time , | |
| 03:33 | just because I want you to understand generally how this | |
| 03:36 | stuff is used in more advanced concepts . So it | |
| 03:39 | all has to do with triangles . This Pythagorean theorem | |
| 03:41 | has to do with triangles , right ? So when | |
| 03:43 | you learn it , you taught you learn about the | |
| 03:44 | concept of what we call a right triangle . A | |
| 03:47 | right triangle is a triangle , any triangle . Uh | |
| 03:50 | that has one special property and that is that one | |
| 03:53 | of the angles is 90°. . So that means that | |
| 03:56 | this is a 90° angle , 90 degree angle means | |
| 04:01 | it goes straight perpendicular like this . So 90 degrees | |
| 04:03 | is exactly like a , like a straight L . | |
| 04:05 | There's no opening up of an angle it straight down | |
| 04:08 | on top of perpendicular to the line under it . | |
| 04:10 | So that's a 90 degree angle right here . Now | |
| 04:12 | when you have a triangle like this , you have | |
| 04:15 | uh side number one , Side , number two , | |
| 04:17 | Side number three , we generally call them side A | |
| 04:20 | side B . And side sea . And side sea | |
| 04:23 | here has to generally be the longest side . So | |
| 04:28 | we kind of label the longest side being C . | |
| 04:31 | A . And B . Doesn't really matter what we | |
| 04:32 | call it , but we always want to use the | |
| 04:33 | variable C . To represent the longest side of the | |
| 04:36 | triangle . We also call this the hypotenuse of the | |
| 04:38 | triangle . I know that you've probably learned that when | |
| 04:40 | you when you studied geometry , you know , years | |
| 04:44 | ago . All right . So the Pythagorean theorem basically | |
| 04:47 | says that if you have a right triangle , it | |
| 04:50 | has to be a right triangle with a 90° angle | |
| 04:52 | . And if the longest side is labeled C , | |
| 04:54 | then it says that the square of that side is | |
| 04:58 | equal to the square of the other side plus the | |
| 05:01 | square of the third side . So C squared is | |
| 05:04 | equal to a squared plus B squared . One of | |
| 05:05 | the most famous formulas . I know you've probably seen | |
| 05:07 | it . If you haven't seen it , that's okay | |
| 05:09 | too . We're gonna go from the very basics here | |
| 05:11 | , but you might look at this and say , | |
| 05:13 | what does this mean ? Right . What it means | |
| 05:15 | is that if I take a right triangle as long | |
| 05:17 | as I have a 90 degree angle here . If | |
| 05:18 | I measure this uh to be 23 centimeters and I | |
| 05:22 | measure this to be some other number of centimeters and | |
| 05:25 | this to be some other number of centimeters . If | |
| 05:27 | I square the length of this side and I square | |
| 05:29 | the length of this side and I add them together | |
| 05:31 | , it should always be equal to the longest side | |
| 05:34 | squared . Now you might look at that and say | |
| 05:36 | how is that true ? How do you know that's | |
| 05:38 | true ? Well , it's kind of like how do | |
| 05:40 | I how do I know I have five fingers on | |
| 05:41 | my hand ? I count them 12345 half five . | |
| 05:44 | You don't really prove ? How do I know I | |
| 05:47 | have five fingers ? You look at it and you | |
| 05:48 | say I have five fingers , how do I know | |
| 05:50 | I have 10 toes ? You know , on both | |
| 05:52 | feet , while I count them , I have 10 | |
| 05:54 | total , right ? It's an observation . How do | |
| 05:56 | we know this is true ? It's because if I | |
| 05:58 | take a ruler and measure this line and measure this | |
| 06:01 | line and measure this line and plug it into this | |
| 06:04 | equation , it actually always equals , No matter if | |
| 06:07 | the triangle is really big , or really , really | |
| 06:09 | small , as long as there's a 90° angle and | |
| 06:11 | there the longest side squared is going to be equal | |
| 06:13 | to the other two sides squared added together . That's | |
| 06:17 | the call the Pythagorean theorem . It holds for all | |
| 06:19 | triangles , but here's the big catch , right ? | |
| 06:23 | We're gonna do a couple of examples here just to | |
| 06:25 | show you . But this pythagorean theorem is only true | |
| 06:28 | . When you draw the triangle on a flat space | |
| 06:31 | , we call it a flat space , which means | |
| 06:33 | this board is flat , right ? So we draw | |
| 06:35 | the triangle on a flat space like this . Then | |
| 06:37 | of course all works in that pythagorean theorem , if | |
| 06:40 | in contrast , and I know this is not a | |
| 06:42 | great glow , but it's a little sphere if instead | |
| 06:46 | I draw those those points of the triangle on a | |
| 06:49 | curved surface . So Einstein talked about curved space and | |
| 06:53 | time . So this is a represent representation of curved | |
| 06:56 | space , curve space time , right . If I | |
| 06:58 | draw one point of the triangle , another point of | |
| 07:00 | the triangle , another point of the triangle . And | |
| 07:02 | I verify it is a right triangle , same as | |
| 07:04 | this . It's just I'm drawing on a curved space | |
| 07:07 | . If I take one side square the other side | |
| 07:09 | squared and then add them together and then compare it | |
| 07:11 | to the longest side squared , then it will not | |
| 07:14 | be equal like this because this pythagorean theorem only holds | |
| 07:18 | in a flat space . It doesn't hold when the | |
| 07:20 | thing is curved , when you draw the triangle on | |
| 07:22 | a curved thing . So you say why is he | |
| 07:24 | telling me this ? Why do I care about that | |
| 07:26 | ? I'm just pointing out that that's a use a | |
| 07:29 | very , very famous use of a very important result | |
| 07:33 | because when you have space and time that are curved | |
| 07:36 | , which was what we call gravity , we call | |
| 07:38 | that . That's what we call the thing that holds | |
| 07:39 | us to the ground . That curvature of space and | |
| 07:42 | time can be measured by how much it doesn't really | |
| 07:46 | work in this equation . Other words , this equation | |
| 07:47 | works for a flat triangle , but if it's very | |
| 07:50 | slightly curved space , then it will be almost equal | |
| 07:53 | . If it's really really curved like a black hole | |
| 07:56 | , then this inequality will be really , really , | |
| 07:59 | really far off . The C . Square will be | |
| 08:01 | totally different than a squared plus B . So the | |
| 08:02 | more curvature you have , the farther away from the | |
| 08:05 | pythagorean theorem , the farther away it doesn't hold anymore | |
| 08:09 | . Right to flatter the space like this chalkboard or | |
| 08:12 | this marker board here , it holds exactly right . | |
| 08:14 | So keep that in the back of your mind as | |
| 08:16 | a use a very famous important use and more advanced | |
| 08:19 | math and science down the road . But for now | |
| 08:21 | let's get back down to reality . Let's take a | |
| 08:24 | look at our triangles . And of course this is | |
| 08:25 | in a flat space . Right ? So let's have | |
| 08:27 | a triangle with three centimeters This direction four centimeters in | |
| 08:30 | this direction and five centimeters This direction . Is this | |
| 08:34 | a right triangle ? Let's check it out . We're | |
| 08:37 | gonna say that A is equal to three and B | |
| 08:39 | is equal to four . And see which is the | |
| 08:41 | longest side . Remember is going to be equal to | |
| 08:43 | five . And we're going to check that out . | |
| 08:45 | We're gonna say , well , is C squared equal | |
| 08:47 | to a squared ? Plus B squared ? Ok , | |
| 08:50 | well , we put see in here we say , | |
| 08:51 | well , five squares on that side . Three squared | |
| 08:54 | goes here . And then four square goes here . | |
| 08:57 | And we're asking ourselves , is it actually equal like | |
| 08:59 | this ? Well , five squared is 25 equals question | |
| 09:03 | mark three times three is nine , and then four | |
| 09:06 | times four is 16 . So then I have what | |
| 09:08 | I have is 25 equals this . When you add | |
| 09:10 | it up is equal to 25 . So because you | |
| 09:13 | can look at this triangle and say three , A | |
| 09:16 | distance of three , a distance of four , a | |
| 09:17 | distance of five works exactly equally . In the Pythagorean | |
| 09:21 | theorem , then you can say with certainty that this | |
| 09:25 | has an actual 90° angle here . Yes , this | |
| 09:29 | is a right triangle . It's a right triangle . | |
| 09:35 | All right now , let's take another triangle . We'll | |
| 09:37 | kind of make room over here . Let's take another | |
| 09:40 | triangle . And and see how it compares . Let's | |
| 09:42 | take another triangle . Let's say we have a triangle | |
| 09:44 | is really long and slender like this . Let's say | |
| 09:48 | this has a distance of one centim , a distance | |
| 09:50 | of nine cm in the distance of 11 cm . | |
| 09:54 | And want to ask ourselves is this also a right | |
| 09:56 | triangle ? Well , in this case A would be | |
| 09:58 | one , B . Would be nine . We don't | |
| 10:00 | care labelling A . And B . These sides , | |
| 10:02 | but we care that the longest side C . Is | |
| 10:04 | always labeled uh with the longest side , which so | |
| 10:07 | C . Is equal to 11 . So again , | |
| 10:09 | we'll say C squared is a squared plus B squared | |
| 10:14 | . We'll ask the question anyway , see square is | |
| 10:16 | going to be 11 squared . We'll ask ourselves , | |
| 10:18 | is that equal to one squared Plus nine Squared ? | |
| 10:22 | Well , 11 times 11 is 121 when you stick | |
| 10:25 | that in your calculator And one times one is one | |
| 10:29 | and nine times nine is 81 . So you can | |
| 10:31 | see the right hand side is going to be 82 | |
| 10:34 | And the left hand side is 121 , so that's | |
| 10:36 | not equal . So because you put these links into | |
| 10:39 | the Pythagorean theorem and it didn't work out there not | |
| 10:41 | equal . Then what you have learned from this is | |
| 10:44 | this is not a 90 degree angle . I've drawn | |
| 10:50 | it close to a 90 degree angle because I'm sketching | |
| 10:52 | it . But if you actually measured one and nine | |
| 10:55 | and connected it with 11 , then you would find | |
| 10:57 | out that this angle is very far away from 90 | |
| 10:59 | degrees . Now , bringing it back to modern thinking | |
| 11:03 | on on science . Okay . We looked at this | |
| 11:05 | first triangle , we said this one fits with the | |
| 11:07 | pythagorean theorem . Why ? Because three squared plus four | |
| 11:10 | squared is equal to 25 which is the longest side | |
| 11:13 | squared . What I was telling you before is that | |
| 11:15 | this relation of the Pythagorean theorem , it only holds | |
| 11:18 | when you draw things in flat space time , flat | |
| 11:21 | space , right ? If I were to take that | |
| 11:23 | globe that I just showed you and mark that triangle | |
| 11:26 | off Exactly three units exactly up straight up four units | |
| 11:30 | exactly over five unit . And try to connect it | |
| 11:32 | , You're going to find it . It's not gonna | |
| 11:33 | work . You're not gonna be able to draw the | |
| 11:35 | triangle on a curved space and have it closed up | |
| 11:39 | on itself like that . Because when you when you | |
| 11:41 | it would be like taking this drawing and trying to | |
| 11:43 | bend it into a sphere , you're gonna distort all | |
| 11:45 | of the distances and angles there . So the Pythagorean | |
| 11:48 | theorem for a 345 triangle , like that's not gonna | |
| 11:51 | work when you curve it . So when we talk | |
| 11:53 | about gravity around a black hole or gravity around the | |
| 11:56 | planet , the space and the time are curved in | |
| 11:58 | such a way that this pythagorean theorem , and later | |
| 12:01 | on , what we're gonna talk about to be , | |
| 12:02 | the distance formula doesn't quite hold in the same way | |
| 12:05 | that it does here . That's how we measure the | |
| 12:06 | curvature of space , is what I'm trying to say | |
| 12:09 | . All right enough , talking about physics , let's | |
| 12:11 | get back into the pure math . Uh so this | |
| 12:14 | is what we call the pythagorean theorem . Now , | |
| 12:16 | we're going to draw a direct extension to what I | |
| 12:18 | know you've probably heard of before , but we're gonna | |
| 12:20 | go a little bit deeper . It's called the distance | |
| 12:23 | formula . Now , most people learn the distance formula | |
| 12:28 | and they just use it because it's not too hard | |
| 12:31 | to use . They don't really know where it comes | |
| 12:32 | from . It turns out that the distance formula is | |
| 12:35 | exactly the same thing as the pythagorean theorem , there | |
| 12:37 | is no difference . And when I say there's no | |
| 12:39 | difference , I mean , literally there is no difference | |
| 12:42 | at all . If you already understand that this is | |
| 12:44 | true , then you already know that . The distance | |
| 12:46 | formula has to be true . And so I want | |
| 12:48 | to show you that rather than just you know , | |
| 12:51 | telling you that and saying believe me , I want | |
| 12:53 | you to understand that . So , let's I want | |
| 12:55 | you to see that . So , let's draw In | |
| 12:57 | order for me to do this , I have to | |
| 12:58 | draw some kind of a coordinate system . So let's | |
| 13:00 | call X . And let's call why some kind of | |
| 13:03 | coordinate system . All right . So in this coordinate | |
| 13:06 | system I'm going to have a couple of points . | |
| 13:08 | I want to find the distance between these two points | |
| 13:10 | . So this point is point number one . I'm | |
| 13:12 | gonna call this P and it's going to be at | |
| 13:15 | coordinates X one , comma Y one . Now , | |
| 13:18 | why am I labelling at X one and Y one | |
| 13:20 | ? Well , because this point can be anywhere . | |
| 13:23 | I'm just putting it right there to illustrate it . | |
| 13:25 | But really the point can be anywhere because I can | |
| 13:27 | measure the distance between any points I want . I'm | |
| 13:29 | just drawing it here . I mean the coordinate here | |
| 13:30 | is probably three comma two or three comma one but | |
| 13:34 | that doesn't matter . It's really at some X coordinate | |
| 13:37 | X . One and Y coordinate Y . One . | |
| 13:39 | That's what that means . And I want to measure | |
| 13:42 | the distance between P . I want to measure it | |
| 13:44 | uh compared to the to another point Q . Which | |
| 13:48 | has some coordinates X . Two comma Y two . | |
| 13:51 | So it's basically point X one , Y . One | |
| 13:54 | and point X two , Y two . Again I've | |
| 13:56 | drawn this thing probably like you know 10 comma nine | |
| 14:00 | or something but doesn't matter . I'm keeping all the | |
| 14:01 | coordinates um basically uh general because it could be anywhere | |
| 14:07 | . Now ultimately I literally want to find the straight | |
| 14:10 | line distance even though I can't draw a perfect straight | |
| 14:12 | line . I'm sorry about that . I guess I | |
| 14:14 | could try a little bit better . I want to | |
| 14:16 | find the straight line distance between these two points . | |
| 14:20 | That's probably not that much better . But you see | |
| 14:21 | if I got a straight edge out I could draw | |
| 14:23 | a straight line between them and say how many centimeters | |
| 14:25 | is it between P and Q . Now there is | |
| 14:28 | a formula that we're going to learn but I want | |
| 14:30 | to show you where the formula comes from . So | |
| 14:32 | what we're gonna do is we're gonna say what is | |
| 14:35 | the actual coordinates ? Let me switch colors here a | |
| 14:38 | little bit . What are the coordinates of this point | |
| 14:40 | ? P . What we already said it's X . | |
| 14:42 | One ? Y . One . Right ? We said | |
| 14:45 | it was X . One Y one . Um Let | |
| 14:47 | me just double check one thing real quick . Yeah | |
| 14:50 | . So if this is X . One , Y | |
| 14:52 | . One basically you can see this coordinate here is | |
| 14:55 | X . One and this coordinate here is why one | |
| 14:58 | , that's what it means to have coordinates X . | |
| 15:00 | One ? Y . One . Right ? And then | |
| 15:02 | this coordinates of Q . Here is X . Two | |
| 15:06 | . And the y coordinates of this point Q over | |
| 15:08 | here is uh we're gonna put this somewhere else uh | |
| 15:12 | Y two . Right ? So I'm gonna put the | |
| 15:16 | white local Y axis label up above like this . | |
| 15:18 | So all I'm doing is showing you this is the | |
| 15:20 | coordinates of this point . This is the coordinate this | |
| 15:22 | point in green , just like this . But if | |
| 15:24 | I really want to find the distance between them to | |
| 15:26 | to make it all come into focus for you , | |
| 15:28 | what I really should draw is the fact that this | |
| 15:30 | forms a triangle . So this forms a base of | |
| 15:33 | the triangle here , and I'm gonna draw this in | |
| 15:36 | blue . I'm not going to cover up the green | |
| 15:37 | to kind of kind of draw in parallel here . | |
| 15:39 | So you can see it kind of forms a right | |
| 15:41 | triangle . Notice this exactly looks like the right triangle | |
| 15:44 | withdrawn right here . You have a longest side called | |
| 15:47 | , see this is the distance between the points we | |
| 15:49 | care about . But there are also these other sides | |
| 15:51 | of the triangle and there's a 90° angle here . | |
| 15:54 | So when you have any two points like this , | |
| 15:56 | it always has a 90° angle like this , and | |
| 15:58 | you can always form a triangle like this . And | |
| 16:01 | the distance between them is the distance that we want | |
| 16:04 | to actually calculate . The distance in this case is | |
| 16:07 | PQ . This is what we want to find the | |
| 16:11 | distance between the PQ . Well , if we learn | |
| 16:14 | from the Pythagorean theorem that the longest side squared is | |
| 16:18 | equal to the other two sides of the triangle squared | |
| 16:20 | and added together , then all we need to do | |
| 16:22 | is figure out what are the other sides of this | |
| 16:24 | triangle here , and we can see it , we | |
| 16:26 | can read it directly from the diagram . What is | |
| 16:28 | the distance of this side of this triangle from here | |
| 16:31 | to here . What would it be ? It's going | |
| 16:33 | to be the point X two minus x one . | |
| 16:35 | Like if this were at nine and this were three | |
| 16:38 | would be nine minus three , that would be the | |
| 16:39 | difference in the in the coordinate . So that would | |
| 16:41 | be the distance here , right ? And then this | |
| 16:44 | distance here is going to be what it's going to | |
| 16:47 | be , Y tu minus Y one because that's the | |
| 16:50 | distance right here between these two points . So this | |
| 16:52 | is why two minus Y one like this . So | |
| 16:57 | if you want to put numbers on it at this | |
| 16:58 | point , we're 10 and this point where why is | |
| 17:01 | equal to 10 and why is equal to to be | |
| 17:02 | 10 -2 ? And you would have eight units here | |
| 17:06 | . So , if you want to find or use | |
| 17:09 | the pythagorean theorem , c squared is a squared plus | |
| 17:11 | B squared . What we're gonna do is we're gonna | |
| 17:13 | say PQ the distance here squared is equal to um | |
| 17:20 | this side here squared X two minus x one squared | |
| 17:26 | plus this distance here squared . Why too ? Minus | |
| 17:30 | y one quantity squared . Make sure you understand what | |
| 17:33 | I'm doing . This is the magic , this is | |
| 17:35 | the secret sauce of what I'm trying to show . | |
| 17:37 | You were gonna end up calculating and finding the distance | |
| 17:40 | formula , which you've probably already seen before , but | |
| 17:42 | I'm showing you that it comes exactly from the pythagorean | |
| 17:44 | theorem . The longest side of this triangle is called | |
| 17:47 | peak . You were saying that distance squared is equal | |
| 17:50 | to this side of the triangle , which is just | |
| 17:52 | X two minus X one squared plus this side of | |
| 17:55 | the triangle , which is just y tu minus Y | |
| 17:57 | one squared . So this is the pythagorean theorem , | |
| 18:00 | C squared is a squared plus b squared . That's | |
| 18:02 | all it is . Now to find the distance . | |
| 18:04 | Of course right now we have PQ squared . So | |
| 18:06 | what we have to do is take the square root | |
| 18:09 | of both sides . So all we'll have is the | |
| 18:11 | distance PQ is equal to , We have to take | |
| 18:15 | the square to both sides . So we take the | |
| 18:16 | square to this side . The square goes away here | |
| 18:19 | we have X two minus x one squared plus y | |
| 18:23 | tu minus y one quantity squared . Now you may | |
| 18:27 | have remembered from algebra uh in the previous lessons that | |
| 18:30 | when you take the square root of both sides , | |
| 18:32 | you have to put a plus or minus in front | |
| 18:33 | of the radical . But what we have here is | |
| 18:35 | we're calculating distances when you have a distance between two | |
| 18:39 | points in the plane , it's always gonna be a | |
| 18:40 | positive number . The distance between me and you is | |
| 18:43 | always going to be a positive number . Even if | |
| 18:45 | I'm going a negative direction , the absolute value of | |
| 18:48 | the distance going in any direction is always positive . | |
| 18:51 | So even though you're always taught to put this plus | |
| 18:53 | minus here , because we're talking about distances , we | |
| 18:56 | never ever need the negative sign . So basically it's | |
| 18:59 | always positive . So because of that , we don't | |
| 19:01 | even need to write the plus or minus at all | |
| 19:02 | . We just say the distance is the square root | |
| 19:05 | of all of this stuff that's under there . Now | |
| 19:07 | in your books , you're probably not gonna see it | |
| 19:09 | written like that , you're gonna see it written like | |
| 19:12 | this , it's going to be called the distance formula | |
| 19:19 | . And what it says is the distance D is | |
| 19:22 | X two minus x one squared plus Y tu minus | |
| 19:28 | y one squared . And then take a nice big | |
| 19:31 | fat square root around the whole entire thing . This | |
| 19:35 | is one of the most important things . This is | |
| 19:36 | probably the central uh concept in this lesson , the | |
| 19:40 | distance formula . So if you want to find the | |
| 19:43 | distance between a point here in the xy plane and | |
| 19:45 | appoint way over there in the xy plane , all | |
| 19:48 | you do is you subtract the x coordinates of the | |
| 19:50 | points and square it . And then you subtract the | |
| 19:53 | y coordinates of the points and square it . You | |
| 19:56 | add those numbers together . And then the last thing | |
| 19:58 | you do is you take a square root again , | |
| 20:00 | most people just use it because it's not that hard | |
| 20:02 | to use , but they don't really know where it | |
| 20:04 | comes from . It comes from the fact that these | |
| 20:06 | things always form triangles , right triangles . And so | |
| 20:09 | it comes from the Pythagorean theorem . This distance formula | |
| 20:12 | is what we used to calculate the distance between points | |
| 20:14 | in space . When you get to more advanced science | |
| 20:17 | , like I told you about gravity , modern theories | |
| 20:19 | of gravity , we don't talk about just space , | |
| 20:21 | we talk about space and time . There is a | |
| 20:23 | very similar equation , not exact , but very close | |
| 20:26 | to this one called the it's called the distance formula | |
| 20:29 | in space time . It's really called the space time | |
| 20:32 | metric really . But it measures the distance between points | |
| 20:34 | in space and time and it looks really close to | |
| 20:37 | this is big . Radical has quantity squared ? It | |
| 20:39 | looks really similar to this . There's a slight change | |
| 20:41 | to it . I don't want to get into it | |
| 20:42 | right now . It has to do with how time | |
| 20:44 | works . But the bottom line is something that you | |
| 20:46 | think is kind of useless actually has far reaching consequences | |
| 20:50 | . So we measure the distance between points in spacetime | |
| 20:53 | . We measure the curvature of gravity by really using | |
| 20:55 | a very similar formula to this uh , with space | |
| 20:59 | and time all mixed together . All right . So | |
| 21:02 | now what we want to do is we want to | |
| 21:03 | use this distance formula to calculate a couple of things | |
| 21:07 | in algebra here , we want to find as an | |
| 21:09 | example the distance between the point negative one comma two | |
| 21:18 | And the .3 comma four . So obviously I could | |
| 21:22 | draw this on a xy plane , I could plot | |
| 21:25 | them and I could draw the triangle . I could | |
| 21:27 | do all the same stuff . I just did . | |
| 21:28 | But ultimately , you don't need to do that anymore | |
| 21:30 | . Once you have the distance formula , there's no | |
| 21:32 | reason to plot it . Every time you can just | |
| 21:35 | put the information directly into the distance formula , knowing | |
| 21:38 | that it always works . So that distance formula is | |
| 21:41 | what again it is X two minus x one squared | |
| 21:46 | plus y two uh minus why ? One quantity squared | |
| 21:51 | . And I'd have to take a square of the | |
| 21:53 | whole thing . Yeah . All right , So now | |
| 21:56 | we have to put the values in here . Now | |
| 21:58 | here's the thing . You have to subtract the values | |
| 22:01 | of X uh X coordinate of one point , an | |
| 22:05 | X coordinate , another point . So let's do it | |
| 22:06 | first one way and show you how it works . | |
| 22:08 | Let's take this is the X coordinate and here's another | |
| 22:11 | X coordinate . So inside of here will say three | |
| 22:14 | minus the minus sign comes from the distance formula . | |
| 22:17 | Then you have a negative one . So you have | |
| 22:19 | a double negative there because it's a minus the minus | |
| 22:21 | one . But then you have to square that and | |
| 22:24 | then you have to go the same way . If | |
| 22:26 | you go from this point subtracting this point , you | |
| 22:28 | have to go to y values in the same direction | |
| 22:30 | , 4 -2 quantity squared . You can't mix up | |
| 22:34 | directions . If you go this way subtracting , you | |
| 22:36 | have to go this way and the other point there | |
| 22:38 | as well . So then you have over here three | |
| 22:42 | minus of minus one is three plus one . So | |
| 22:44 | you have four squared and then over here four minutes | |
| 22:46 | to is two squared . And so you have the | |
| 22:48 | square root of all of that . So the distance | |
| 22:50 | between these points is 16 plus four , Which is | |
| 22:55 | the square root of 20 . And so you have | |
| 22:57 | to ask yourself what is the square root of 20 | |
| 22:59 | ? Well , I can do a factor tree here | |
| 23:01 | , right ? I can do 10 times too And | |
| 23:04 | five times too . So I can have a circle | |
| 23:06 | of pair here . So what I'm going to get | |
| 23:08 | is the distance this too comes out of the radical | |
| 23:12 | square root of five , two square to five and | |
| 23:15 | that's the final answer . So you say what is | |
| 23:17 | two square to five means ? Well it means if | |
| 23:20 | I grab a sheet of paper and put xy tick | |
| 23:23 | marks on it , let's say I measured it in | |
| 23:25 | meters or centimeter . Whatever the units you pick is | |
| 23:27 | , what the unit of the answer is going to | |
| 23:28 | be . If I put the first coordinate negative one | |
| 23:31 | centimeter and then up two centimeters and I put this | |
| 23:34 | one at three centimeters and up four centimeters then the | |
| 23:37 | distance if I measured it with a ruler between those | |
| 23:40 | points with literally putting a ruler between them would be | |
| 23:43 | to times square to five . Now you can put | |
| 23:44 | this in a calculator and get the decimal , you | |
| 23:46 | could get some value and decimal but that would be | |
| 23:49 | in centimeters . If you put the original points in | |
| 23:51 | terms of meters then the answer you get for the | |
| 23:54 | distance would be in meters . If the points were | |
| 23:56 | in terms of light years , then the answer you | |
| 23:59 | get would be light years . You see , it | |
| 24:00 | doesn't it doesn't matter whatever units you use for the | |
| 24:03 | points is going to give you and dictate the units | |
| 24:05 | that you get in the answer between the two points | |
| 24:08 | . Now , one more important thing I want to | |
| 24:10 | point out is that it doesn't matter which point is | |
| 24:14 | X two and which point is X one in this | |
| 24:16 | case I did three minus two minus one . And | |
| 24:18 | then because of that I did four minus two to | |
| 24:20 | but it doesn't matter which point is X one in | |
| 24:23 | which point is X two but you just have to | |
| 24:25 | be consistent . For instance let's go the other direction | |
| 24:28 | . So let's say instead of going calling this X | |
| 24:31 | two in this X one we'll flip it around and | |
| 24:33 | say this is X two and this is X one | |
| 24:35 | . So we'll go the other direct the other way | |
| 24:37 | , we'll say negative one minus three , subtracting this | |
| 24:40 | direction squared . Then if we do it this direction | |
| 24:44 | we have to be consistent . So we have to | |
| 24:45 | do to minus four quantity squared . I want you | |
| 24:49 | to make sure that you understand that this is exactly | |
| 24:51 | backwards from what we did here . The negative three | |
| 24:53 | minus the negative one is exactly backwards from negative one | |
| 24:57 | minus three . And then the four minus two is | |
| 25:00 | exactly backwards of the tu minus four . But we're | |
| 25:02 | gonna get the same answer because what do we have | |
| 25:04 | here ? Negative one minus three is negative four squared | |
| 25:08 | to minus four is negative two squared . And you | |
| 25:11 | can see that the negatives , it's not gonna matter | |
| 25:13 | because everything is squared inside , you're still gonna have | |
| 25:16 | the 16 , you're still going to have the four | |
| 25:18 | , you're still going to have the squared of 20 | |
| 25:20 | and so you're still going to have to times square | |
| 25:22 | to five . So the most important thing to realize | |
| 25:25 | for the distance formula is when you're calculating distances , | |
| 25:29 | it does not matter which direction you subtract , but | |
| 25:33 | you must be consistent if you pick a point and | |
| 25:35 | say this minus this for X , then you must | |
| 25:37 | also pick the same direction for why ? When you're | |
| 25:40 | doing the subtraction . So , we covered a lot | |
| 25:44 | so far , we've covered the pythagorean theorem , We've | |
| 25:47 | shown you that when you draw these triangles on a | |
| 25:49 | flat board like this anyway , And it has a | |
| 25:52 | right 90° angle here , then the pythagorean holes uh | |
| 25:57 | , squared is a squared plus B squared . We | |
| 25:59 | showed you that the distance formula comes from the Pythagorean | |
| 26:02 | theorem . So this distance formula you get will hold | |
| 26:05 | for any points . Again , in flat space , | |
| 26:08 | we're not talking about black holes or gravity or neutron | |
| 26:10 | stars . We're talking about on a chalkboard on a | |
| 26:12 | sheet of paper and we calculated the distance between two | |
| 26:15 | points . And we showed you that it doesn't matter | |
| 26:17 | the direction used to do this attraction . So we're | |
| 26:19 | going to do more problems . But that's the general | |
| 26:21 | idea . Now , the last thing we want to | |
| 26:23 | talk about is something called the midpoint formula . Some | |
| 26:26 | of you have been exposed to this and some of | |
| 26:28 | you have it , but it's really , really simple | |
| 26:29 | to understand . What we want to do is if | |
| 26:32 | we have two points , like we did in the | |
| 26:33 | last part , what we're finding the distance between them | |
| 26:36 | , let's say we don't care about the distance . | |
| 26:38 | We just want to figure out if we have two | |
| 26:39 | points in space , where is the point between them | |
| 26:43 | ? Now ? Of course I'm holding my fingers up | |
| 26:44 | . So you know , the point is somewhere here | |
| 26:46 | in the middle between them . But what I mean | |
| 26:47 | by where is the point I'm talking about ? If | |
| 26:50 | I give you the coordinates of two end points and | |
| 26:53 | you can put your finger in the middle of cutting | |
| 26:56 | the thing in half . Where is that point ? | |
| 26:58 | In terms of what are its coordinates ? That's called | |
| 27:00 | the midpoint of the two original points you have . | |
| 27:04 | And so we have something called the midpoint formula . | |
| 27:06 | Now again , I don't want to just blab it | |
| 27:09 | out for you . I want you to understand where | |
| 27:11 | it comes from . So we have something called the | |
| 27:13 | midpoint formula . It's very easy to understand . It's | |
| 27:21 | actually easier to understand than uh any of the other | |
| 27:25 | guys here . So what we have here is let's | |
| 27:27 | go ahead and again draw an X . Y plane | |
| 27:32 | , it's not going to be perfect . So this | |
| 27:33 | is X . And this is why and same kind | |
| 27:35 | of thing . I'm gonna draw to random points here | |
| 27:37 | , P and Q . So I'm going to call | |
| 27:39 | this again , P this is X one comma Y | |
| 27:43 | one exactly the same before . And this thing we're | |
| 27:45 | going to call it Q . It's going to be | |
| 27:47 | at some coordinates , X two , comma Y two | |
| 27:50 | . Right now , I know that there's a straight | |
| 27:53 | line that that connects these guys . I mean , | |
| 27:56 | I've drawn that , I know how to calculate that | |
| 27:58 | that's covered with the distance formula . Right ? But | |
| 28:01 | I don't want to actually figure out the distance between | |
| 28:04 | them . Let's say I want to figure out where | |
| 28:07 | exactly is the midpoint of this line segments ? Probably | |
| 28:10 | somewhere around around there . It's hard for me to | |
| 28:11 | tell . But there's a point somewhere here that's exactly | |
| 28:14 | midway between the two end points . Like if this | |
| 28:16 | were five centimeters and this were five centimeters and the | |
| 28:19 | whole thing will be 10 centimetres . It's right in | |
| 28:20 | the middle . I want to figure out what is | |
| 28:22 | the coordinates of this thing ? Right ? How do | |
| 28:25 | I figure that out ? Well , first let's go | |
| 28:28 | take a look at point . P . What are | |
| 28:30 | the coordinates of P already talked about this before ? | |
| 28:32 | This is X . one And over here is why | |
| 28:35 | one . Right now we have some coordinate point Q | |
| 28:40 | . And its coordinates are X . Two . And | |
| 28:43 | the y coordinates of this point is why sub tube | |
| 28:47 | nothing has changed from before ? Everything is exactly the | |
| 28:49 | same . So my question to you is how can | |
| 28:52 | I figure out what this point is in the middle | |
| 28:54 | in terms of its coordinates ? How could I possibly | |
| 28:57 | figure that out ? Well , this point has to | |
| 29:00 | have some kind of X value and it has to | |
| 29:02 | have some kind of why value . And the way | |
| 29:06 | you figure out what the midpoint is is you kind | |
| 29:08 | of forget about why for a second you just look | |
| 29:11 | kind of , if you could just look down from | |
| 29:12 | above then you would say the endpoint has some point | |
| 29:15 | along X here and the endpoint has some point along | |
| 29:18 | X here . So midway between this thing has to | |
| 29:22 | be equal distance from here to here and from here | |
| 29:24 | to here . So this point right here , the | |
| 29:25 | X value of it has to be the average X | |
| 29:29 | two minus X 1/2 . In other words , the | |
| 29:32 | value of this X coordinate of the midpoint is the | |
| 29:35 | average of X two and X one . In other | |
| 29:38 | words , I mean if you think about it , | |
| 29:40 | if if I have the the endpoint , the X | |
| 29:42 | coordinate of the point , is that two ? Or | |
| 29:44 | let's make it easy if the endpoint is at zero | |
| 29:47 | and the other end point is at 10 . I'm | |
| 29:49 | talking along the X axis , then we know half | |
| 29:51 | way , it's gotta be at five . So 10 | |
| 29:53 | plus zero divided by two is five . You're just | |
| 29:56 | averaging the two points . If you pick any two | |
| 29:58 | points , you want to find the middle of it | |
| 30:00 | , you take the average , that's what you do | |
| 30:01 | , right ? So to find the middle in the | |
| 30:03 | X direction , you just average the X coordinates and | |
| 30:06 | the exact same thing is gonna happen over here . | |
| 30:08 | The y value here is going to be y tu | |
| 30:11 | minus Y . One over to you . Just average | |
| 30:13 | the Y values . So if I give you a | |
| 30:16 | point P and a point Q . You can always | |
| 30:18 | tell me the pinpoint , the midpoint is going to | |
| 30:20 | have an average of the X . Values for the | |
| 30:21 | X coordinate and an average of the Y . Values | |
| 30:24 | for the Y coordinate . So the midpoint of the | |
| 30:32 | segment joining P . located at X one , Y | |
| 30:41 | 1 and Q . Located at X . To comment | |
| 30:46 | Y two is this is the way it's written in | |
| 30:50 | a text book and it's really confusing the way it's | |
| 30:51 | written a lot of times but this is the way | |
| 30:53 | it's usually written Mm represents the midpoint . The x | |
| 30:56 | coordinate of that midpoint is X one plus X 2/2 | |
| 31:02 | . That's the average of the X values . The | |
| 31:04 | y coordinate is the average of the y values . | |
| 31:09 | Let's put it just do it like this . Why | |
| 31:10 | ? One plus Y 2/2 . You see all this | |
| 31:13 | is saying it looks really confusing but all that's basically | |
| 31:15 | saying is the midpoint has an X . Value of | |
| 31:18 | the average of the x coordinates . And the y | |
| 31:20 | value has the average of the y coordinates . So | |
| 31:24 | it's gonna be easier to show with an example right | |
| 31:27 | , what is the midpoint between the segment defined by | |
| 31:39 | uh four comma negative six and negative three comma two | |
| 31:45 | . Now , of course I could plot for common | |
| 31:47 | , negative six and I could plot negative three comma | |
| 31:49 | too . And I could put my finger in the | |
| 31:51 | middle and say ha ha it's about right there and | |
| 31:53 | do all that . But I don't need to do | |
| 31:54 | that . I mean I have the midpoint formula , | |
| 31:56 | I know what it says . And so the way | |
| 32:00 | to do this is you say OK , the X | |
| 32:02 | value of the midpoint is just going to be the | |
| 32:05 | average of these x values here . So it's gonna | |
| 32:08 | be four plus the negative three over to four minus | |
| 32:14 | three is going to give you one and then you're | |
| 32:15 | gonna have a two . So the X value of | |
| 32:18 | the midpoint there is just at a location of one | |
| 32:21 | half and then you're gonna have the why value of | |
| 32:25 | the midpoint ? Which is again the average of these | |
| 32:28 | guys here , negative six plus two , negative six | |
| 32:30 | plus 2/2 . That's how you average things . Right | |
| 32:33 | , So you're gonna get on top , you're gonna | |
| 32:35 | get a negative 4/2 and then you're gonna get a | |
| 32:37 | negative two for this . So what you would write | |
| 32:40 | down for your final answer is the midpoint exactly between | |
| 32:48 | these two , join the line segment , joining these | |
| 32:51 | two points is gonna have an X coordinate of one | |
| 32:53 | half and a Y coordinate of negative too . And | |
| 32:55 | I promise you , if you get some graph paper | |
| 32:57 | out and you plot this point , you plot at | |
| 32:58 | this point and you you look at this point , | |
| 33:01 | it's gonna be right in the middle of the segment | |
| 33:03 | , just like this . All right . So that | |
| 33:06 | was a long lesson . I had to kind of | |
| 33:07 | cover it all together because in the next few lessons | |
| 33:10 | , we're gonna have topics that uh kind of jumble | |
| 33:13 | all of these concepts together . So you have to | |
| 33:15 | know what each of them all is . And so | |
| 33:17 | we can do some more complicated problems . Pythagorean theorem | |
| 33:20 | is something that is not proven , it is something | |
| 33:23 | that is just observed any time you draw a triangle | |
| 33:26 | with a right 90 degree angle in one corner , | |
| 33:29 | then you always know that this relation holds if you | |
| 33:32 | labelled along the side C . And the other two | |
| 33:34 | side , it doesn't really matter then when you this | |
| 33:36 | equality holds again , I showed you that in curved | |
| 33:39 | space . I mean , I didn't really show you | |
| 33:41 | , but I'm telling you that in curved space this | |
| 33:43 | relationship doesn't hold the angles inside the triangle also get | |
| 33:46 | distorted and looked weird as well . And then when | |
| 33:49 | we look at the distance formula , you can see | |
| 33:51 | it comes directly from the pythagorean theorem . We did | |
| 33:54 | problems to calculate the distance between points . We talked | |
| 33:56 | about it doesn't matter which direction you do the subtraction | |
| 33:59 | , as long as you're consistent . When you pick | |
| 34:01 | one direction , you have to pick the other the | |
| 34:03 | same direction for the Y value . And then we | |
| 34:05 | talked about the concept of midpoint between two points and | |
| 34:08 | the fact that it's just an average of the X | |
| 34:10 | coordinates and the Y coordinates . So I want you | |
| 34:12 | to make sure you understand these concepts . Follow me | |
| 34:15 | on to the next lesson . We're gonna do some | |
| 34:16 | more complicated problems dealing with the pythagorean there in the | |
| 34:19 | midpoint formula and the distance formula . |
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