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In this video we're going to get introduced to the

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Pythagorean theorem, which is fun on its own.

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But you'll see as you learn more and more mathematics it's

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one of those cornerstone theorems of really all of math.

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It's useful in geometry, it's kind of the backbone

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of trigonometry.

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You're also going to use it to calculate distances

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between points.

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So it's a good thing to really make sure we know well.

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So enough talk on my end.

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Let me tell you what the Pythagorean theorem is.

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So if we have a triangle, and the triangle has to be a right

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triangle, which means that one of the three angles in the

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triangle have to be 90 degrees.

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And you specify that it's 90 degrees by drawing that

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little box right there.

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So that right there is-- let me do this in a different

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color-- a 90 degree angle.

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Or, we could call it a right angle.

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And a triangle that has a right angle in it is

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called a right triangle.

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So this is called a right triangle.

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Now, with the Pythagorean theorem, if we know two sides

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of a right triangle we can always figure out

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the third side.

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And before I show you how to do that, let me give you one

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more piece of terminology.

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The longest side of a right triangle is the side opposite

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the 90 degree angle-- or opposite the right angle.

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So in this case it is this side right here.

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This is the longest side.

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And the way to figure out where that right triangle is, and

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kind of it opens into that longest side.

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That longest side is called the hypotenuse.

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And it's good to know, because we'll keep referring to it.

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And just so we always are good at identifying the hypotenuse,

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let me draw a couple of more right triangles.

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So let's say I have a triangle that looks like that.

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Let me draw it a little bit nicer.

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So let's say I have a triangle that looks like that.

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And I were to tell you that this angle right

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here is 90 degrees.

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In this situation this is the hypotenuse, because it is

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opposite the 90 degree angle.

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It is the longest side.

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Let me do one more, just so that we're good at

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recognizing the hypotenuse.

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So let's say that that is my triangle, and this is the 90

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degree angle right there.

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And I think you know how to do this already.

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You go right what it opens into.

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That is the hypotenuse.

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That is the longest side.

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So once you have identified the hypotenuse-- and let's say

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that that has length C.

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And now we're going to learn what the Pythagorean

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theorem tells us.

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So let's say that C is equal to the length of the hypotenuse.

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So let's call this C-- that side is C.

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Let's call this side right over here A.

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And let's call this side over here B.

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So the Pythagorean theorem tells us that A squared-- so

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the length of one of the shorter sides squared-- plus

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the length of the other shorter side squared is going to

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be equal to the length of the hypotenuse squared.

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Now let's do that with an actual problem, and you'll see

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that it's actually not so bad.

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So let's say that I have a triangle that looks like this.

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Let me draw it.

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Let's say this is my triangle.

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It looks something like this.

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And let's say that they tell us that this is the right angle.

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That this length right here-- let me do this in different

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colors-- this length right here is 3, and that this

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length right here is 4.

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And they want us to figure out that length right there.

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Now the first thing you want to do, before you even apply the

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Pythagorean theorem, is to make sure you have your

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hypotenuse straight.

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You make sure you know what you're solving for.

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And in this circumstance we're solving for the hypotenuse.

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And we know that because this side over here, it is the side

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opposite the right angle.

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If we look at the Pythagorean theorem, this is C.

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This is the longest side.

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So now we're ready to apply the Pythagorean theorem.

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It tells us that 4 squared-- one of the shorter sides-- plus

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3 squared-- the square of another of the shorter sides--

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is going to be equal to this longer side squared-- the

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hypotenuse squared-- is going to be equal to C squared.

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And then you just solve for C.

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So 4 squared is the same thing as 4 times 4.

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That is 16.

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And 3 squared is the same thing as 3 times 3.

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So that is 9.

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And that is going to be equal to C squared.

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Now what is 16 plus 9?

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It's 25.

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So 25 is equal to C squared.

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And we could take the positive square root of both sides.

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I guess, just if you look at it mathematically, it could

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be negative 5 as well.

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But we're dealing with distances, so we only care

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about the positive roots.

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So you take the principal root of both sides and

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you get 5 is equal to C.

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Or, the length of the longest side is equal to 5.

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Now, you can use the Pythagorean theorem, if we give

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you two of the sides, to figure out the third side no matter

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what the third side is.

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So let's do another one right over here.

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Let's say that our triangle looks like this.

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And that is our right angle.

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Let's say this side over here has length 12, and let's say

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that this side over here has length 6.

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And we want to figure out this length right over there.

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Now, like I said, the first thing you want to do is

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identify the hypotenuse.

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And that's going to be the side opposite the right angle.

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We have the right angle here.

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You go opposite the right angle.

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The longest side, the hypotenuse, is right there.

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So if we think about the Pythagorean theorem-- that A

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squared plus B squared is equal to C squared-- 12

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you could view as C.

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This is the hypotenuse.

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The C squared is the hypotenuse squared.

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So you could say 12 is equal to C.

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And then we could say that these sides, it doesn't matter

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whether you call one of them A or one of them B.

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So let's just call this side right here.

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Let's say A is equal to 6.

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And then we say B-- this colored B-- is equal

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to question mark.

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And now we can apply the Pythagorean theorem.

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A squared, which is 6 squared, plus the unknown B squared is

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equal to the hypotenuse squared-- is equal

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to C squared.

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Is equal to 12 squared.

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And now we can solve for B.

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And notice the difference here.

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Now we're not solving for the hypotenuse.

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We're solving for one of the shorter sides.

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In the last example we solved for the hypotenuse.

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We solved for C.

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So that's why it's always important to recognize that A

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squared plus B squared plus C squared, C is the length

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of the hypotenuse.

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So let's just solve for B here.

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So we get 6 squared is 36, plus B squared, is equal

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to 12 squared-- this 12 times 12-- is 144.

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Now we can subtract 36 from both sides of this equation.

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Those cancel out.

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On the left-hand side we're left with just a B squared

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is equal to-- now 144 minus 36 is what?

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144 minus 30 is 114.

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And then you subtract 6, is 108.

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So this is going to be 108.

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So that's what B squared is, and now we want to take the

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principal root, or the positive root, of both sides.

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And you get B is equal to the square root, the

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principal root, of 108.

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Now let's see if we can simplify this a little bit.

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The square root of 108.

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And what we could do is we could take the prime

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factorization of 108 and see how we can

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simplify this radical.

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So 108 is the same thing as 2 times 54, which is the same

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thing as 2 times 27, which is the same thing as 3 times 9.

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So we have the square root of 108 is the same thing as the

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square root of 2 times 2 times-- well actually,

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I'm not done.

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9 can be factorized into 3 times 3.

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So it's 2 times 2 times 3 times 3 times 3.

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And so, we have a couple of perfect squares in here.

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Let me rewrite it a little bit neater.

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And this is all an exercise in simplifying radicals that you

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will bump into a lot while doing the Pythagorean theorem,

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so it doesn't hurt to do it right here.

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So this is the same thing as the square root of 2 times 2

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times 3 times 3 times the square root of that last

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3 right over there.

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And this is the same thing.

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And, you know, you wouldn't have to do all of

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this on paper.

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You could do it in your head.

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What is this?

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2 times 2 is 4.

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4 times 9, this is 36.

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So this is the square root of 36 times the square root of 3.

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The principal root of 36 is 6.

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So this simplifies to 6 square roots of 3.

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So the length of B, you could write it as the square root of

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108, or you could say it's equal to 6 times the

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square root of 3.

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This is 12, this is 6.

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And the square root of 3, well this is going to be a 1

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point something something.

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So it's going to be a little bit larger than 6.

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