Garfield's Pythagorean Theorem Proof

Date: 02/27/97 at 23:10:00
From: Mr. Michael D. Ferris, Sr.
Subject: Pythagorean Theorem

Dear Mr. Math:

I am a ninth grader at Tupelo High School (Tupelo, Mississippi), and 
my Accelerated Geometry Teacher has assigned the following question 
for extra credit:

Which President of the United States wrote a proof of the 
Pythagorean Theorem? On what did he base his proof?  Give examples and 
provide copy of proof.

We have searched the Internet and the library at school and have 
been unable to find information concerning these questions.

Thank you so much!

Date: 02/28/97 at 12:14:57
From: Doctor Wilkinson
Subject: Re: Pythagorean Theorem

The president was James Garfield, and this is the proof:

Take two copies of your right triangle with sides a, b, c.  Place 
one of them with side b horizontal and side a vertical with the right 
angle at the lower left.  Place the second triangle with its side b 
as a continuation of the side a of the first triangle and with the 
right angle at the upper left.  Join the other vertices of the two 
triangles.  This give you a trapezoid with upper base a, lower base b,
and height a + b.  Now we're going to compute the area of the 
trapezoid in two different ways.

First we have the formula

        A = h * (a + b) / 2

In this case h = a + b, so we get

        A = (a + b)^2 / 2

On the other hand, the trapezoid is made up of three triangles. Two
of these are the two copies of the original right triange. The third
is the isosceles right triangle with two sides equal to c. The sum
of the areas of these is

        (1/2) ab + (1/2) ab + (1/2) c^2

If you equate the two formulas you get the Pythagorean Theorem.

Pretty clever, isn't it?

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/