Video Transcript: Pythagorean Theorem

Intro to the Pythagorean theorem 

In this video we're going to get introduced to the Pythagorean theorem, which is fun on its  own. But you'll see as you learn more and more mathematics it's one of those cornerstone  theorems of really all of math. It's useful in geometry, it's kind of the backbone of trigonometry. You're also going to use it to calculate distances between points. So it's a good thing to really  make sure we know well. So enough talk on my end. Let me tell you what the Pythagorean  theorem is. So if we have a triangle, and the triangle has to be a right triangle, which means  that one of the three angles in the triangle have to be 90 degrees. And you specify that it's 90  degrees by drawing that little box right there. So that right there is-- let me do this in a  different color-- a 90 degree angle. Or, we could call it a right angle. And a triangle that has a  right angle in it is called a right triangle. So this is called a right triangle. Now, with the  Pythagorean theorem, if we know two sides of a right triangle we can always figure out the  third side. And before I show you how to do that, let me give you one more piece of  terminology. The longest side of a right triangle is the side opposite the 90 degree angle-- or  opposite the right angle. So in this case it is this side right here. This is the longest side. And the  way to figure out where that right triangle is, and kind of it opens into that longest side. That  longest side is called the hypotenuse. And it's good to know, because we'll keep referring to it.  And just so we always are good at identifying the hypotenuse, let me draw a couple of more  right triangles. So let's say I have a triangle that looks like that. Let me draw it a little bit nicer.  So let's say I have a triangle that looks like that. And I were to tell you that this angle right here  is 90 degrees. In this situation this is the hypotenuse, because it is opposite the 90 degree  angle. It is the longest side. Let me do one more, just so that we're good at recognizing the  hypotenuse. So let's say that that is my triangle, and this is the 90 degree angle right there. And I think you know how to do this already. You go right what it opens into. That is the hypotenuse. That is the longest side. So once you have identified the hypotenuse-- and let's say that that  has length C. And now we're going to learn what the Pythagorean theorem tells us. So let's say  that C is equal to the length of the hypotenuse. So let's call this C-- that side is C. Let's call this  side right over here A. And let's call this side over here B. So the Pythagorean theorem tells us  that A squared-- so the length of one of the shorter sides squared-- plus the length of the other  shorter side squared is going to be equal to the length of the hypotenuse squared. Now let's do  that with an actual problem, and you'll see that it's actually not so bad. So let's say that I have a  triangle that looks like this. Let me draw it. Let's say this is my triangle. It looks something like  this. And let's say that they tell us that this is the right angle. That this length right here-- let me do this in different colors-- this length right here is 3, and that this length right here is 4. And  they want us to figure out that length right there. Now the first thing you want to do, before  you even apply the Pythagorean theorem, is to make sure you have your hypotenuse straight.  You make sure you know what you're solving for. And in this circumstance we're solving for the  hypotenuse. And we know that because this side over here, it is the side opposite the right  angle. If we look at the Pythagorean theorem, this is C. This is the longest side. So now we're  ready to apply the Pythagorean theorem. It tells us that 4 squared-- one of the shorter sides--  plus 3 squared-- the square of another of the shorter sides-- is going to be equal to this longer  side squared-- the hypotenuse squared-- is going to be equal to C squared. And then you just  solve for C. So 4 squared is the same thing as 4 times 4. That is 16. And 3 squared is the same 

thing as 3 times 3. So that is 9. And that is going to be equal to C squared. Now what is 16 plus  9? It's 25. So 25 is equal to C squared. And we could take the positive square root of both sides. I guess, just if you look at it mathematically, it could be negative 5 as well. But we're dealing with distances, so we only care about the positive roots. So you take the principal root of both sides  and you get 5 is equal to C. Or, the length of the longest side is equal to 5. Now, you can use the  Pythagorean theorem, if we give you two of the sides, to figure out the third side no matter  what the third side is. So let's do another one right over here. Let's say that our triangle looks  like this. And that is our right angle. Let's say this side over here has length 12, and let's say that this side over here has length 6. And we want to figure out this length right over there. Now,  like I said, the first thing you want to do is identify the hypotenuse. And that's going to be the  side opposite the right angle. We have the right angle here. You go opposite the right angle.  The longest side, the hypotenuse, is right there. So if we think about the Pythagorean  theorem-- that A squared plus B squared is equal to C squared-- 12 you could view as C. This is  the hypotenuse. The C squared is the hypotenuse squared. So you could say 12 is equal to C.  And then we could say that these sides, it doesn't matter whether you call one of them A or one of them B. So let's just call this side right here. Let's say A is equal to 6. And then we say B-- this  colored B-- is equal to question mark. And now we can apply the Pythagorean theorem. A  squared, which is 6 squared, plus the unknown B squared is equal to the hypotenuse squared--  is equal to C squared. Is equal to 12 squared. And now we can solve for B. And notice the  difference here. Now we're not solving for the hypotenuse. We're solving for one of the shorter  sides. In the last example we solved for the hypotenuse. We solved for C. So that's why it's  always important to recognize that A squared plus B squared plus C squared, C is the length of  the hypotenuse. So let's just solve for B here. So we get 6 squared is 36, plus B squared, is equal  to 12 squared-- this 12 times 12-- is 144. Now we can subtract 36 from both sides of this  equation. Those cancel out. On the left-hand side we're left with just a B squared is equal to--  now 144 minus 36 is what? 144 minus 30 is 114. And then you subtract 6, is 108. So this is going  to be 108. So that's what B squared is, and now we want to take the principal root, or the  positive root, of both sides. And you get B is equal to the square root, the principal root, of 108.  Now let's see if we can simplify this a little bit. The square root of 108. And what we could do is  we could take the prime factorization of 108 and see how we can simplify this radical. So 108 is  the same thing as 2 times 54, which is the same thing as 2 times 27, which is the same thing as 3 times 9. So we have the square root of 108 is the same thing as the square root of 2 times 2  times-- well actually, I'm not done. 9 can be factorized into 3 times 3. So it's 2 times 2 times 3  times 3 times 3. And so, we have a couple of perfect squares in here. Let me rewrite it a little bit  neater. And this is all an exercise in simplifying radicals that you will bump into a lot while doing the Pythagorean theorem, so it doesn't hurt to do it right here. So this is the same thing as the  square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there. And this is the same thing. And, you know, you wouldn't have to do all of this on paper. You could do it in your head. What is this? 2 times 2 is 4. 4 times 9, this is 36. So this is the square root of 36  times the square root of 3. The principal root of 36 is 6. So this simplifies to 6 square roots of 3.  So the length of B, you could write it as the square root of 108, or you could say it's equal to 6  times the square root of 3. This is 12, this is 6. And the square root of 3, well this is going to be a  1 point something something. So it's going to be a little bit larger than 6.

Pythagorean theorem example 

Say we have a right triangle. Let me draw my right triangle just like that. This is a right triangle.  This is the 90 degree angle right here. And we're told that this side's length right here is 14. This side's length right over here is 9. And we're told that this side is a. And we need to find the  length of a. So as I mentioned already, this is a right triangle. And we know that if we have a  right triangle, if we know two of the sides, we can always figure out a third side using the  Pythagorean theorem. And what the Pythagorean theorem tells us is that the sum of the  squares of the shorter sides is going to be equal to the square of the longer side, or the square  of the hypotenuse. And if you're not sure about that, you're probably thinking, hey Sal, how do  I know that a is shorter than this side over here? How do I know it's not 15 or 16? And the way to tell is that the longest side in a right triangle, and this only applies to a right triangle, is the side  opposite the 90 degree angle. And in this case, 14 is opposite the 90 degrees. This 90 degree  angle kind of opens into this longest side. The side that we call the hypotenuse. So now that we know that that's the longest side, let me color code it. So this is the longest side. This is one of  the shorter sides. And this is the other of the shorter sides. The Pythagorean theorem tells us  that the sum of the squares of the shorter sides, so a squared plus 9 squared is going to be  equal to 14 squared. And it's really important that you realize that it's not 9 squared plus 14  squared is going to be equal to a squared. a squared is one of the shorter sides. The sum of the  squares of these two sides are going to be equal to 14 squared, the hypotenuse squared. And  from here, we just have to solve for a. So we get a squared plus 81 is equal to 14 squared. In  case we don't know what that is, let's just multiply it out. 14 times 14. 4 times 4 is 16. 4 times 1  is 4 plus 1 is 5. Take a 0 there. 1 times 4 is 4. 1 times 1 is 1. 6 plus 0 is 6. 5 plus 4 is 9, bring down  the 1. It's 196. So a squared plus 81 is equal to 14 squared, which is 196. Then we could subtract  81 from both sides of this equation. On the left-hand side, we're going to be left with just the a  squared. These two guys cancel out, the whole point of subtracting 81. So we're left with a  squared is equal to 196 minus 81. What is that? If you just subtract 1, it's 195. If you subtract 80,  it would be 115 if I'm doing that right. And then to solve for a, we just take the square root of  both sides, the principal square root, the positive square root of both sides of this equation. So  let's do that. Because we're dealing with distances, you can't have a negative square root, or a  negative distance here. And we get a is equal to the square root of 115. Let's see if we can break down 115 any further. So let's see. It's clearly divisible by 5. If you factor it out, it's 5, and then 5  goes in the 115 23 times. So both of these are prime numbers. So we're done. So you actually  can't factor this anymore. So a is just going to be equal to the square root of 115. Now if you  want to get a sense of roughly how large the square root of 115 is, if you think about it, the  square root of 100 is equal to 10. And the square root of 121 is equal to 11. So this value right  here is going to be someplace in between 10 and 11, which makes sense if you think about it  visually. 

Pythagorean theorem with isosceles triangle 

We're asked to find the value of x in the isosceles triangle shown below. So that is the base of  this triangle. So pause this video and see if you can figure that out. Well the key realization to  solve this is to realize that this altitude that they dropped, this is going to form a right angle  here and a right angle here and notice, both of these triangles, because this whole thing is an 

isosceles triangle, we're going to have two angles that are the same. This angle, is the same as  that angle. Because it's an isosceles triangle, this 90 degrees is the same as that 90 degrees.  And so the third angle needs to be the same. So that is going to be the same as that right over  there. And since you have two angles that are the same and you have a side between them that is the same this altitude of 12 is on both triangles, we know that both of these triangles are  congruent. So they're both going to have 13 they're going to have one side that's 13, one side  that is 12 and so this and this side are going to be the same. So this is going to be x over two  and this is going to be x over two. And so now we can use that information and the fact and the  Pythagorean Theorem to solve for x. Let's use the Pythagorean Theorem on this right triangle  on the right hand side. We can say that x over two squared that's the base right over here this  side right over here. We can write that x over two squared plus the other side plus 12 squared is  going to be equal to our hypotenuse squared. Is going to be equal to 13 squared. This is just the  Pythagorean Theorem now. And so we can simplify. This is going to be x. We'll give that the  same color. This is going to be x squared over four. That's just x squared over two squared plus  144 144 is equal to 13 squared is 169. Now I can subtract 144 from both sides. I'm gonna try to  solve for x. That's the whole goal here. So subtracting 144 from both sides and what do we get?  On the left hand side, we have x squared over four is equal to 169 minus 144. Let's see, 69  minus 44 is 25. So this is going to be equal to 25. We can multiply both sides by four to isolate  the x squared. And so we get x squared is equal to 25 times four is equal to 100. Now, if you're  just looking this purely mathematically and say, x could be positive or negative 10. But since  we're dealing with distances, we know that we want the positive value of it. So x is equal to the  principle root of 100 which is equal to positive 10. So there you have it. We have solved for x.  This distance right here, the whole thing, the whole thing is going to be equal to 10. Half of that is going to be five. So if we just looked at this length over here. I'm doing that in the same  column, let me see. So this length right over here, that's going to be five and indeed, five  squared plus 12 squared, that's 25 plus 144 is 169, 13 squared. So the key of realization here is  isosceles triangle, the altitudes splits it into two congruent right triangles and so it also splits  this base into two. So this is x over two and this is x over two. And we use that information and  the Pythagorean Theorem to solve for x.