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Welcome to the presentation on the Pythagorean theorem. I apologize if my voice sounds a little horsie. A little hoarse, not horsie. I was singing a little bit too much last night, so please forgive me. Well, anyway, we will now teach you about the Pythagorean theorem. And you might have heard of this before. As far as I know, it is the only mathematical theorem named after the founder of a religion. Pythagoras, actually, I think his whole religion was based on mathematics. But I'm no historian here. So I'll leave that to the historians. But anyway, let's get started on what the Pythagorean theorem is all about. If I were to give you a triangle-- let me give you a triangle-- and I were to tell you that it's not a normal triangle. It is a right triangle. And all a right triangle is is a triangle that has one side equal to 90 degrees. And I'll leave you to think about whether it's ever possible for a triangle to have more than one side that's 90 degrees. But anyway, just granted that a right triangle is a side that has at least-- well, let me say a right triangle is a triangle that has only one side that's at 90 degrees. And if you have a right triangle, what the Pythagorean theorem allows you to do is if I give you a right triangle and I give you two of the sides, we can figure out the third side. So before I throw the theorem at you, let me actually give you a couple of more definitions. Actually, just one more. So if this is the right angle in a right triangle-- it's at 90 degrees. And we symbolize that by drawing the angles like this, kind of like a box instead of drawing it like a curve, like that. I hope I'm not messing up the drawing too much. The side opposite the right angle is called the hypotenuse. And I really should look up where this word comes from. Because I think it's a large and unwieldy word, and it's a little daunting at first. My sister told me that she had a math teacher once who made people memorize it's a high pot that is in use. So I don't know if that helps you or not. But over time, you'll use the term hypotenuse so much that it'll seem just like a normal word. Although when you look at it, it really does look kind of strange. Anyway, going back to definitions, the hypotenuse is the side opposite the 90-degree angle. And if you look at any right triangle, you'll also quickly realize that the hypotenuse is the longest side of the right triangle. So I think we're done now with definitions. So what does the Pythagorean theorem tell us? Well, let's call C is equal to the length of the hypotenuse. And let A be the length of this side. And let B equal the length of this side. What the Pythagorean theorem tells us is that A squared plus B squared is equal to C squared. Now, that very simple formula might be one of the most powerful formulas in mathematics. From this, you go into Euclidean geometry. You go into trigonometry. You can do anything with this formula, but we'll leave that to future lectures. Let's actually test this formula. Or not test it-- let's use the formula. Maybe in another presentation, I'll actually do a proof or, at minimum, a visual proof of it. I apologize ahead of time that I'm a bit scatterbrained today. It's been a while since I last did a video. And once again, I told you I sang a little bit too much last night. So my throat is sore. OK, so we have a triangle. And remember, it has to be a right triangle. So let's say that this is a right angle right here. It's 90 degrees. And if I were to tell you that this side is of length 4. And actually, let me change that. This side is of length 3. This side is of length 4. And we want to figure out the side of this length. So the first thing I do when I look at a right triangle is I figure out what the hypotenuse is. Which side is the hypotenuse? Well, there's two ways to do it. There's actually one way. You look at where the right angle is. And it's the side opposite to that. So this is the hypotenuse. This would be C in our formula, the Pythagorean theorem. We could call it whatever we want. But just for simplicity, remember A squared plus B squared is equal to C squared. So in this case, we see that the other two sides, each of them squared, when added together will equal C squared. So we get 3 squared plus 4 squared is equal to C squared, where C is our hypotenuse. So 3 squared is 9, plus 16 is equal to C squared. 25 is equal to C squared. And C could be plus or minus 5. But we know that you can't have a minus 5 length in geometry. So we know that C is equal to 5. So using the Pythagorean theorem, we just figured out that if we know the sides-- if one side is 3, the other side is 4, then we can use Pythagorean theorem to figure out that the hypotenuse of this triangle has the length 5. Let's do another example. Let's say, once again, this is the right angle. This side is of length 12. This slide is of length 6. And I want to figure out what this side is. So let's write down the Pythagorean theorem. A squared plus B squared is equal to C squared, where C is that length of the hypotenuse. So the first thing I want to do when I look at our triangle that I just drew is which side is the hypotenuse. Well, this right here is the right angle. So the hypotenuse is this side right here. And we can also eyeball it and say, oh, that's definitely the longest side of this triangle. So we know that A squared plus B squared is equal to 12 squared, which is 144. Now we know we have one side, but we don't have the other side. So I've got to ask you a question. Does it matter which side we substitute for A or B? Well, no, just because A or B-- they kind of do the same thing in this formula. So we could pick any side to be A other than the hypotenuse. And we'll pick the other side to be B. So let's just say that this side is B, and let's say that this side is A. So we know what A is. So we get 6 squared plus B squared is equal to 144. So we get 36 plus B squared is equal to 144. B squared is equal to 144 minus 36. B squared is equal to 112. Now we've got to simplify what the square root of 112 is. And what we did in those radical modules probably is helpful here. So let's see. B is equal to the square root of 112. Let's think about it. How many times does 4 go into 112? 4 goes into 120 five times, so it'll go into it 28 times. And then 4 goes into 28 seven times. So I actually think that this is equal to 16 times 7. Am I right? 7 times 10 is 70, plus 42 is 112. Right. So B equals the square root of 16 times 7. See, I just factored that as a product of a perfect square and a prime number. Or actually, it doesn't have to be a prime number, just a non-perfect square. And then I get B is equal to 4 square roots of 7. So there we go. If this is 12, this is 6, this is 4 square roots of 7. I think that's all the time I have now for this presentation. Right after this, I'll do one more presentation where I give a couple of more Pythagorean theorem problems. See you soon.