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## Pythagorean theorem proofs

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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.