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Garfield's proof of the Pythagorean theorem

Former U.S. President James Garfield wrote a proof of the Pythagorean theorem. He used a trapezoid made of two identical right triangles and half of a square to show that the sum of the squares of the two shorter sides equals the square of the longest side of a right triangle. Created by Sal Khan.

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  • spunky sam blue style avatar for user Mark Bray
    what is theta
    (35 votes)
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  • piceratops tree style avatar for user Sebastian Newell
    I'm reading through the slightly different answers people all gave to the different people who asked the same question- "what is theta?". Well, my question is what is the most commonly used definition for theta?
    (26 votes)
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  • piceratops sapling style avatar for user Saralun Ondee
    At , what do you mean by "multiply out" (a+b) square and get a square + 2ab + b square?
    (37 votes)
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    • aqualine ultimate style avatar for user Megh
      Yeah, thats why (a+b)^2 always equals to a^2 + 2ab + b^2, it is an algebraic exponent property.
      For Example: When a = 2 and b = 3,
      (2+3)^2 = 2^2 + 2(2 x 3) + 3^2
      5^2 = 4 + 2 x 6 + 9
      25 = 13 + 12
      25 = 25
      As L.H.S. = R.H.S. : (a+b)^2 = a^2 + 2ab + b^2
      (10 votes)
  • aqualine ultimate style avatar for user L.H.Marten
    How is it posable for someone to come up with a theorem that is right? how was he able to make this up suddenly? A squrd x B squrd = C squrd, how can someone come up with something like this?
    (14 votes)
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    • piceratops ultimate style avatar for user D S
      Theorems aren't right they are just scaffolding which is used to build up more theorems. For this they only need to be consistent, which is different from being right or true or fact or real or any of these other categories. As for how thoughts become theorems, it's a process of de-simplification. Which isn't ever just made up, but is built upon other ideas then articulated. Flying squirrels and ants perform trigonometry, but humans talk about it and put it in language. Humans 40,000 years ago started articulating and thinking and then 4,000 years ago, which is when written history starts, we see they were turning thought into theory, and since then it's just been a matter of building up scaffolding, in different languages. One being algebra which started being used to express this ancient relationship of the subtend to the arms (of a triangle but also of a square and a rectangle as the ancient Chinese, Indians, and Babylonian theories went) after the Middle Ages when Italian and German and others discovered ancient Greek and current Arabic/Hindu math, formed groups and schools of method and adopted symbols and abstractions instead of writing everything out in Latin. It was also around that time, coinciding with the rise of professional mathematicians, when simply 'learning for learnings sake' or 'contributing to the stores of wisdom in God's House', sadly gave way to the modern idea of 'originality' and 'research' which has a completely different ethic about it. But it seems Mr. Garfield here wasn't hindered and took the time to light upon an ancient idea even though it had been looked at by countless eyes before.
      (27 votes)
  • aqualine seed style avatar for user shreyasbhat2001
    why do we need to equate the area of trapezoid to the are of the three triangles??
    (10 votes)
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    • blobby green style avatar for user ramanvesh
      Mr. Garfield realized that there are two ways of calculating the area, and it is understood that, since we are calculating the area of the same figure, both those methods SHOULD give the same result. So he just tried to calculate using both those ways, and saw what it gave him. This is a common way of proof in mathematics - U know something through method A, and you know it using method B also. And since you know that both these, should be giving the same result, by equating them, you might stumble upon wonderful new things :). That is the magic of mathematics.

      Beware that sometimes, you may end up with something obvious like a.b = a.b, but sometimes, you might end up proving Pythagoras theorem or that e^i(pi) + 1= 0. :D
      (22 votes)
  • female robot ada style avatar for user Terrylee Brewer
    At how does he get the 2ab by multiplying (a+b) squared out? I don't understand. Shouldn't it just be a squared plus b squared?
    (4 votes)
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    • duskpin ultimate style avatar for user James Brown
      Work it out with real numbers to see how it comes out.
      Let a = 3
      Let b = 5
      So we start with (a + b)^2
      Plugging in the numbers, we'd have (3 + 5)^2 or 8-squared which is equal to 64.

      But if we presume that (a + b)^2 is equal to a-squared + b-squared, and if we plug in the numbers, we'd have 3-squared ( which is 9) plus 5-squared (which is 25) and added together they make only 34. Not even close to 64.

      On the other hand, we see that a^2 + 2ab + b^2 when substituted would work out to:
      3-squared + 2(3 times 5) + 5-squared, or 9 + 2(15) + 25, which is equal to 9 + 30 + 25 = 64.
      (13 votes)
  • female robot grace style avatar for user M. Wancewicz
    Was James Garfield the most recent to prove the Pythagorean Theorem in a new way?
    (15 votes)
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  • primosaur ultimate style avatar for user Swift Runner
    Is Ө just a symbol, or does it have mathematical meaning?
    (6 votes)
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  • starky ultimate style avatar for user Kat
    Did anyone understand this? I am very confuzzled.
    (8 votes)
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  • blobby blue style avatar for user Forrest Li
    Why (A+B) squared is equal to A squared + 2AB + B squared?
    (2 votes)
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    • duskpin tree style avatar for user Glorfindel
      First, try to change the expression (A+B)^2 into something manageable and understandable. (A+B)^2 = (A+B) * (A+B). Using the Distributive Property, (A+B) * (A+B) = (A+B) * A + (A+B) * B. To further simplify the expression, (A+B) * A + (A+B) * B = A^2 + AB + AB + B^2, which is equal to A^2 + 2AB + B^2.

      '^' is the same as raised to the nth power. '*' is the same as multiplied with.

      Hope this helps!
      (8 votes)

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.