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Garfield's proof of the Pythagorean theorem


Video transcript

What we're going to do in this video is study a proof of the Pythagorean theorem that was first discovered, or as far as we know first discovered, by James Garfield in 1876, and what's exciting about this is he was not a professional mathematician. You might know James Garfield as the 20th president of the United States. He was elected president. He was elected in 1880, and then he became president in 1881. And he did this proof while he was a sitting member of the United States House of Representatives. And what's exciting about that is that it shows that Abraham Lincoln was not the only US politician or not the only US President who was really into geometry. And what Garfield realized is, if you construct a right triangle-- so I'm going to do my best attempt to construct one. So let me construct one right here. So let's say this side right over here is length b. Let's say this side is length a, and let's say that this side, the hypotenuse of my right triangle, has length c. So I've just constructed enough a right triangle, and let me make it clear. It is a right triangle. He essentially flipped and rotated this right triangle to construct another one that is congruent to the first one. So let me construct that. So we're going to have length b, and it's collinear with length a. It's along the same line, I should say. They don't overlap with each other. So this is side of length b, and then you have a side of length-- let me draw a it so this will be a little bit taller-- side of length b. And then, you have your side of length a at a right angle. Your side of length a comes in at a right angle. And then, you have your side of length c. So the first thing we need to think about is what's the angle between these two sides? What's this mystery angle? What's that mystery angle going to be? Well, it looks like something, but let's see if we can prove to ourselves that it really is what we think it looks like. If we look at this original triangle and we call this angle "theta," what's this angle over here, the angle that's between sides of length a and length c? What's the measure of this angle going to be? Well, theta plus this angle have to add up to 90. Because you add those two together, they add up to 90. And then, you have another 90. You're going to get 180 degrees for the interior angles of this triangle. So these two have to add up to 90. This angle is going to be 90 minus theta. Well, if this triangle appears congruent-- and we've constructed it so it is congruent-- the corresponding angle to this one is this angle right over here. So this is also going to be theta, and this right over here is going to be 90 minus theta. So given that this is theta, this is 90 minus theta, what is our angle going to be? Well, they all collectively go 180 degrees. So you have theta, plus 90 minus theta, plus our mystery angle is going to be equal to 180 degrees. The thetas cancel out. Theta minus theta. And you have 90 plus our mystery angle is 180 degrees. Subtract 90 from both sides, and you are left with your mystery angle equaling 90 degrees. So that all worked out well. So let me make that clear, and that's going to be useful for us in a second. It's going to be useful. So we can now say definitively that this is 90 degrees. This is a right angle. Now, what we are going to do is we are going to construct a trapezoid. This side a is parallel to side b down here, the way it's been constructed, and this is just one side right over here. This goes straight up, and now let's just connect these two sides right over there. So there's a couple of ways to think about the area of this trapezoid. One is we could just think of it as a trapezoid and come up with its area, and then we could think about it as the sum of the areas of its components. So let's just first think of it as a trapezoid so what do we know about the area of a trapezoid? Well, the area of a trapezoid is going to be the height of the trapezoid, which is a plus b. That's the height of the trapezoid. Times-- the way I think of it-- the mean of the top and the bottom, or the average of the top and the bottom. Since that's this times one half times a plus a plus b. And the intuition there, you're taking the height times the average of this bottom and the top. The average of the bottom and the top gives you the area of the trapezoid. Now, how could we also figure out the area with its component parts? Regardless of how we calculate the area, as long as we do correct things, we should come up with the same result. So how else can we come up with this area? Well, we could say it's the area of the two right triangles. The area of each of them is one half a times b, but there's two of them. Let me do that b in that same blue color. But there's two of these right triangle. So let's multiply by two. So two times one half ab. That takes into consideration this bottom right triangle and this top one. And what's the area of this large one that I will color in in green? What's the area of this large one? Well, that's pretty straightforward. It's just one half c times c. So plus one half c times c, which is one half c squared. Now, let's simplify this thing and see what we come up with, and you might guess where all of this is going. So let's see what we get. So we can rearrange this. Let me rearrange this. So one half times a plus b squared is going to be equal to 2 times one half. Well, that's just going to be one. So it's going to be equal to a times b, plus one half c squared. Well, I don't like these one halfs laying around, so let's multiply both sides of this equation by 2. I'm just going to multiply both sides of this equation by 2. On the left-hand side, I'm just left with a plus b squared. So let me write that. And on the right-hand side, I am left with 2ab. Trying to keep the color coding right. And then, 2 times one half c squared, that's just going to be c squared plus c squared. Well, what happens if you multiply out a plus b times a plus b? What is a plus b squared? Well, it's going to be a squared plus 2ab plus 2ab plus b squared. And then, our right-hand side it's going to be equal to all of this business. And changing all the colors is difficult for me, so let me copy and let me paste it. So it's still going to be equal to the right-hand side. Well, this is interesting. How can we simplify this? Is there anything that we can subtract from both sides? Well, sure there is. You have a 2ab on the left-hand side. You have a 2ab on the right-hand side. Let's subtract 2ab from both sides. If you subtract 2ab from both sides, what are you left with? You are left with the Pythagorean theorem. So you're left with a squared plus b squared is equal to c squared. Very, very exciting. And for that, we have to thank the 20th president of the United States, James Garfield. This is really exciting. The Pythagorean theorem, it was around for thousands of years before James Garfield, and he was able to contribute just kind of fiddling around while he was a member of the US House of Representatives.