Math Antics - The Pythagorean Theorem - By Mathantics
Transcript
00:03 | Uh huh . Hi , I'm rob . Welcome to | |
00:07 | Math Antics . In this lesson , We're going to | |
00:09 | learn about the Pythagorean theorem or Pythagoras theorem as it's | |
00:13 | sometimes called . And you may be wondering what's the | |
00:15 | theorem and who in the world is Pythagoras ? Well | |
00:20 | , in math , a theorem is simply a statement | |
00:22 | that has been proven to be true from other things | |
00:25 | that are either known or accepted to be true . | |
00:27 | And Pythagoras . Well , he was this really smart | |
00:30 | dude who lived a long time ago in ancient Greece | |
00:32 | and he proved the theorem . Well , historians aren't | |
00:35 | completely sure it was actually Pythagoras who proved it . | |
00:38 | It could have been one of his students or followers | |
00:41 | , but he usually gets credit for it anyway . | |
00:46 | The main thing that you need to know is that | |
00:47 | the Pythagorean theorem describes an important geometric relationship between the | |
00:52 | three sides of a right triangle . We're going to | |
00:54 | learn what that relationship is in just a minute . | |
00:57 | But first , there's several things that you need to | |
00:59 | know before you can truly understand the Pythagorean theorem or | |
01:02 | use it to solve problems . First of all , | |
01:04 | to understand the pythagorean theorem , you need to know | |
01:07 | about angles and triangles , and you also need to | |
01:09 | know a little bit about exponents and square roots . | |
01:12 | So if those topics are new to you , be | |
01:14 | sure to watch our videos about them . First second | |
01:17 | , even though the pythagorean theorem is about geometry , | |
01:20 | you'll need to know some basic algebra to actually use | |
01:22 | it specifically . You'll need to know about variables and | |
01:25 | how to solve basic algebraic equations that involve exponents . | |
01:28 | We cover a lot of those topics in the first | |
01:31 | five videos of our algebra basic series . Okay , | |
01:35 | now that you've got all that background info covered , | |
01:37 | let's see what the Pythagorean theorem actually says . The | |
01:40 | theorem can be stated in several different ways , but | |
01:42 | the one we like best goes like this for a | |
01:45 | right triangle with legs A and B . And hip | |
01:47 | . Until you see a squared plus B squared equals | |
01:51 | C . Squared . As you can see from this | |
01:53 | definition , the pythagorean theorem doesn't apply to all triangles | |
01:57 | , It only applies to right triangles , as you | |
02:00 | know , right triangles always include one right angle that's | |
02:03 | usually marked with a square right angle symbol . To | |
02:06 | help you identify it , and you need to know | |
02:09 | which angle is the right angle because it helps you | |
02:11 | identify an important side of the triangle called the hypotenuse | |
02:15 | . The hypotenuse is the longest side of a right | |
02:17 | triangle , and it's always the side that's opposite of | |
02:20 | the right angle . In other words , it's the | |
02:22 | side that doesn't touch or help form the right angle | |
02:25 | itself . In order to use the pythagorean theorem , | |
02:29 | you need to be able to identify the hypotenuse because | |
02:31 | that's what the variable C . Stands for . In | |
02:33 | the theorem , C . Is the length of the | |
02:35 | hypotenuse side . The other two sides of the triangle | |
02:39 | . The ones that do touch or form the right | |
02:41 | angle are called its legs . Our pythagorean theorem definition | |
02:45 | uses the variable names A and B . To represent | |
02:48 | their lengths . Oh , and it doesn't matter which | |
02:50 | leg is called A . And which leg is called | |
02:52 | B . As long as you keep track of which | |
02:54 | is which after you make your initial choice . Okay | |
02:58 | . Now that we know the various parts of the | |
03:00 | Pythagorean theorem , let's think about what the relationship or | |
03:03 | equation A squared plus B squared equals C squared is | |
03:07 | really telling us it's telling us that if we take | |
03:10 | the lengths of the two legs sides A and B | |
03:13 | . And square them , which means multiplying them by | |
03:16 | themselves . A squared is eight times a and B | |
03:19 | squared as B times B . And then if we | |
03:22 | add those two squared amounts together , they will equal | |
03:26 | the amount you'd get if you square the hypotenuse side | |
03:28 | , which would be C squared or C . Time | |
03:31 | . See that may sound a little confusing at first | |
03:34 | . So let's take a look at a special example | |
03:36 | of a right triangle . That will help the pythagorean | |
03:38 | theorem make a little more sense . This right triangle | |
03:41 | is called a 345 triangle because its sides have the | |
03:44 | relative lengths of 34 and five and by relative lengths | |
03:49 | , I mean that the units of length don't really | |
03:51 | matter . Besides . Could be expressed in any units | |
03:54 | inches , meters miles , whatever . So the triangle | |
03:58 | could be of any size as long as it's lengths | |
04:01 | would have the proportions 34 and five relative to each | |
04:04 | other , starting with the side . That's three units | |
04:07 | long . Which will call side A . What do | |
04:10 | we get if we square that side ? Well , | |
04:12 | an arithmetic squaring three means multiplying three times three which | |
04:16 | equals nine . And the geometric equivalent of squaring something | |
04:20 | actually results in a square shape . As you can | |
04:23 | see . This square contains nine unit squares , so | |
04:26 | this red area represents the value a squared in the | |
04:29 | Pythagorean theorem . Next , let's look at the side | |
04:32 | that's four units long , which will call site B | |
04:35 | . Squaring four means multiplying four times 4 which is | |
04:38 | 16 . Again , the geometric equivalent of that is | |
04:41 | a literal square . That is four units on each | |
04:44 | side and covers a total area of 16 units . | |
04:47 | So this blue area represents B squared in the Pythagorean | |
04:50 | theorem . And finally , let's deal with the hypotenuse | |
04:54 | or sight see which is the longest side . It's | |
04:56 | five units long . Squaring five means multiplying five times | |
05:00 | 5 which is 25 . And the geometric equivalent is | |
05:04 | a five x 5 square . That has an area | |
05:06 | of 25 units . So , this green area represents | |
05:09 | c squared in the Pythagorean theorem . Now that you | |
05:12 | can see how the arithmetic parts of the Pythagorean theorem | |
05:15 | are related to the geometric parts of this right triangle | |
05:18 | . Let's check to see if the pythagorean theorem is | |
05:20 | really true , at least in this special case on | |
05:23 | the arithmetic side , if you add up the amounts | |
05:25 | A squared and B squared , they really do equals | |
05:28 | C squared because nine plus 16 equals 25 . And | |
05:32 | with a little rearranging of our unit squares , you | |
05:34 | can see that the area of the squares formed by | |
05:37 | the two legs really does equal the area of the | |
05:39 | square form by the hypotenuse , wow ! Those ancient | |
05:42 | greek dudes really were smart . Okay . But I | |
05:46 | know what some of you are thinking . That's cool | |
05:48 | and all . But why should I even care about | |
05:50 | the pythagorean theorem ? What's it good for ? Well | |
05:53 | , as always , that's a good question . And | |
05:55 | the answer is like many things in math . The | |
05:58 | pythagorean theorem is a useful tool that can help you | |
06:01 | use what you do know to figure out what you | |
06:03 | don't know specifically if you have a right triangle , | |
06:06 | but you only know how long two of its sights | |
06:08 | are . The Pythagorean theorem tells you how to figure | |
06:11 | out the length of the third , unknown side . | |
06:13 | For example , imagine that you have a right triangle | |
06:16 | that's two cm long on this side and three cm | |
06:19 | long on this side . But we don't know how | |
06:21 | long the hypothesis . No problemo , the Pythagorean theorem | |
06:25 | tells us the relationship between all three sides of any | |
06:28 | right triangle . So we can figure it out . | |
06:31 | We know that a squared plus B squared equals C | |
06:34 | squared . So let's plug in what we do know | |
06:36 | into that equation and then solve it for what we | |
06:38 | don't know . Again it doesn't matter which of the | |
06:41 | two legs is called A . Or B . So | |
06:43 | let's just label them like this and then substitute to | |
06:45 | for A . And three for B . And the | |
06:48 | pythagorean theorem equation . That gives us an algebraic equation | |
06:52 | that has just one unknown sea . If we solve | |
06:55 | this equation for C . In other words , if | |
06:57 | we rearrange the equation so that C . Is all | |
06:59 | by itself on one side of the equal sign , | |
07:02 | then we'll know exactly what C is . Well , | |
07:04 | no , the length of that side of the triangle | |
07:07 | First . We need to simplify the left side of | |
07:09 | the equation since it contains the known numbers . And | |
07:12 | according to the order of operations , we need to | |
07:14 | simplify the exponents . 1st two squared is four and | |
07:18 | three squared is nine . Then we add those results | |
07:21 | four plus nine equals 13 and we have the equation | |
07:24 | 13 equals c squared which is the same as c | |
07:27 | squared equals 13 . Then to get see all by | |
07:30 | itself , we need to do the inverse of what's | |
07:32 | being done to it since it's being squared , the | |
07:35 | inverse operation is the square root . So we need | |
07:38 | to take the square root of both sides . Taking | |
07:41 | the square root of c squared just gives us C | |
07:44 | . Which is what we want on this side of | |
07:45 | the equation . But it gives us a little problem | |
07:48 | on the other side because it's not easy to figure | |
07:50 | out what the square root of 13 is . It's | |
07:52 | not a perfect square . So it's going to be | |
07:54 | a decimal and probably an international number but that's okay | |
07:58 | because it's fine to just leave our answer as the | |
08:00 | square root of 13 . Sure you could use a | |
08:03 | calculator to get the decimal value if you really need | |
08:05 | one . But in math it's very common to just | |
08:08 | leave square roots alone . Unless they're easy to simplify | |
08:11 | . So the sides of this right triangle are two | |
08:14 | centimeters three centimeters and the square root of 13 centimeters | |
08:19 | . Let's try another example for this right triangle . | |
08:21 | We know the length of the hypotenuse 6m and one | |
08:25 | of the legs 4m but the length of the other | |
08:27 | leg is unknown . So let's use the Pythagorean theorem | |
08:30 | to find that unknown length . As usual we call | |
08:34 | the hypotenuse side C . And let's call the leg | |
08:36 | , we know site A and the leg we don't | |
08:38 | know side B . Then we can substitute the known | |
08:42 | values into the Pythagorean theorem and solve for the unknown | |
08:44 | value , replacing the C with six . And the | |
08:48 | A . With four gives us the equation four squared | |
08:51 | plus B squared equals six squared , which we need | |
08:55 | to simplify and solve for B first let's simplify the | |
08:58 | exponents four squared is 16 and six squared is 36 | |
09:03 | . Now we need to isolate the B squared and | |
09:06 | we can do that by subtracting 16 from both sides | |
09:09 | of the equation On this side the Plus 16 and | |
09:12 | the -16 . Leave us with just be squared . | |
09:15 | And on the other side we have 36 -16 which | |
09:18 | is 20 . We can now solve the simplified equation | |
09:22 | for B by taking the square root of both sides | |
09:24 | , which gives us B equals the square root of | |
09:27 | 20 . Again it's fine to leave your answer as | |
09:30 | a square root like this and some of you may | |
09:32 | know that the square root of 20 can be simplified | |
09:35 | to two times the square root of five . We're | |
09:37 | not going to worry about simplifying roots in this video | |
09:40 | , but if you know how to do it awesome | |
09:42 | . If you don't know , just leave the answer | |
09:44 | as the square root of 20 m . Here's another | |
09:47 | interesting one . What if you have a unit square | |
09:49 | that's cut in half along a diagonal . Each side | |
09:52 | of the square is one unit long , But how | |
09:55 | far is it from one corner of the square to | |
09:57 | the other along the diagonal ? Well , since the | |
10:00 | diagonal divides the square into two right triangles , we | |
10:03 | can use the pythagorean theorem to tell us that unknown | |
10:06 | distance . We labeled the legs of the right triangle | |
10:09 | A . And B . And the hypotenuse C . | |
10:12 | And since we know that A and B are both | |
10:13 | one , we can plug those values into the pythagorean | |
10:16 | theorem equation , which gives us one squared plus one | |
10:19 | squared equals c squared . Now we solve for C | |
10:23 | . One squared is just one . So the left | |
10:25 | side of this equation simplifies to one plus one , | |
10:28 | which is just two . That means c squared equals | |
10:31 | two . And if we take the square root of | |
10:33 | both sides , we get C equals the square root | |
10:36 | of two . So that's how far it is across | |
10:38 | the diagonal of the unit square . Okay , so | |
10:42 | that's how you use the pythagorean theorem to find the | |
10:44 | length of an unknown side of a right triangle , | |
10:46 | which is its most common use . But there's another | |
10:49 | way that you can use the Pythagorean theorem that I | |
10:51 | want to mention . You can also use the Pythagorean | |
10:54 | theorem to test a triangle to see if it truly | |
10:57 | is a right triangle . You know , in case | |
10:59 | you're not already sure for example what if someone shows | |
11:03 | you this triangle ? and ask , is this a | |
11:05 | right triangle ? Well , it looks a lot like | |
11:07 | a right triangle , but it doesn't have a right | |
11:09 | angle symbol . And it would be kind of hard | |
11:11 | to tell if this angle is exactly 90 degrees just | |
11:14 | by looking at it . Maybe it's really close to | |
11:16 | 90 like 89.5 degrees . No worries . That the | |
11:20 | factory and theorem can tell us for sure . If | |
11:22 | we know the lengths of all three sides of the | |
11:24 | triangle , if we know those lengths A , B | |
11:27 | and C , then we can just plug them into | |
11:29 | the pythagorean theorem equation to see if it holds true | |
11:32 | in this particular case , since the two shorter sides | |
11:35 | are each three centimeters and the longest side is four | |
11:37 | centimeters . We can plug those values in for A | |
11:40 | . B and C . And simplify to see what | |
11:42 | we get three squared is nine . So on this | |
11:45 | side of the equation , we get nine plus nine | |
11:47 | which is 18 . And on the other side we | |
11:50 | have four squared which is 16 . Oh , that | |
11:53 | doesn't look right . Our equations simplified to 18 equals | |
11:57 | 16 , which is definitely not a true statement . | |
12:01 | That means that the three sides of this triangle do | |
12:03 | not work in the pythagorean theorem . They don't fit | |
12:05 | the relationship A squared plus B squared equals c squared | |
12:09 | . And since the pythagorean theorem tells us that all | |
12:12 | right triangles , that that relationship , this triangle must | |
12:15 | not be a right triangle . All right . So | |
12:18 | now , you know what the Pythagorean theorem is , | |
12:20 | and you know how to use it , you can | |
12:22 | use it to find a missing side of any right | |
12:24 | triangle . And you can also use it to test | |
12:27 | the triangle to see if it qualifies as a right | |
12:29 | triangle . But as you can see , it takes | |
12:32 | a lot of other mass skills to be able to | |
12:33 | use the Pythagorean theorem effectively . So you may need | |
12:36 | to brush up on some of those skills before you're | |
12:39 | ready to try using it on your own . And | |
12:41 | remember you can't get good at math just by watching | |
12:43 | videos about it . You actually need to practice solving | |
12:46 | real math problems , as always . Thanks for watching | |
12:48 | Math Antics and I'll see you next time learn more | |
12:52 | at Math antics dot com . |
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