Lesson 6: Finding side lengths of triangles
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.