Garfield's proof of the Pythagorean Theorem

James Garfield's proof of the Pythagorean Theorem.

What we're doing in this video is study a proof of the Pythagorean theorem,

that was first discovered,as far as we know by James Garfield in 1876.

What's exciting about this is that he was not a professional mathematician.

You might know James Garfield as the twentieth president of the United States.

He was elected president 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.

What's exciting about that is is it shows that Abraham Lincoln was not the only US politician

or the only US president who was into geometry. And what Garfield realised is that we can construct a right triangle-

Let's say this side over here is length 'b'(blue) and this side is length 'a'(red),

and let's say this side, the hypotenuse of my right triangle, has length 'c'.

And let me make it clear -- it is a right triangle. He essentially flipped and rotated this 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 colinear with length 'a', It's along the same line as length 'a.'

They don't overlap with each other.

So this is a side of length 'b.'
And then you have your side of length 'a'

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 going to be?

Well, it looks like something, but let's see if we can prove

to ourselves 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 the sides of length a and c.

what's the measure of this angle going to be?

Well, theta plus this angle has to add up to 90, because when you add those two together, they add up to the 90.

So 90 and 90 you get 180 degrees for the interior angles of this triangle.

So if these two angles together is 90, then this angle is '90 minus theta'.

Well if this angle up here is congruent -- (And we've constructed it so it is congruent.) the angle corresponding to theta is also going to be theta,

And this angle right over here is also going to be 90 - theta.

So given that this is theta and this is 90 - theta, what is our angle going to be?

Well they all, collectively, add up to 180 degrees.

So you have theta + (90-theta) + our mystery angle is going to be equal to 180 degrees.

The thetas cancel out (theta - theta), and you have 90 + our mystery angle is a 180 degrees,

We subtract 90 from both sides.) - and you are left with

your mystery angle equalling 90 degrees.

So that all worked out well.

So let me make that clear. That's going to be useful for us.

So now we can 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 its 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 can 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 trapezoid.

So, what do we know about the area of the trapezoid?

The area of a trapezoid,is gonna be the height of the trapezoid, which is (a+b) times, the way I think of it, the mean or average of the top and the bottom.

So, ar(trapezoid) = (a+b) x 1/2(a+b)

In the intuition there you are taking the height times the average of the bottom and the top, gives you the area of the trapezoid.

Now, how can we also figure out the area with its component parts?

So as far as we do the 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 of a times b.

But there's two of them, Let me do that say in blue colour,

But there's two of these right triangles, so let's multiply them by two.

So 2 times 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'll colour in green

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 square.

Now , let's simplify this thing and see what we come up with and you might guess where all of this is going.

So, we can rearrange this. So this one half times (a+b) squared is going to be equal to two times one half,

well that's just going to be one, so its gonna be equal to a times b plus one half c squared.

I don't like these one halves lying around, so let's multiply both sides, this equation, by 2.

I'm just gonna multiply both sides by two.

So, on the left hand side, I'm just left with (a+b) squared,

and on the right hand side, I'm left with 2ab and then two times one half c squared, i.e., plus c squared.

What happens if you multiply out (a+b) times (a+b)? We get (a+b) squared.

That is a sq. + 2ab+ b sq. = 2ab + c sq.

Subtracting 2ab from both sides, we are left with, a sq. + b sq. = c sq. i.e. the Pythagorean theorem.