## President Garfield

American president James A. Garfield devised an original proof of the Pythagorean theorem using the following figure. Can you complete the proof?

Source: NCTM Mathematics Teacher

SOLUTION
Area of trapezoid = $2\times$ area of right triangle $abc\;+$ area of right triangle $cc$
Area of trapezoid = half the product of the height and the sum of the bases
Area of triangle = half the product of a base and the height to that base
$\frac{1}{2}\left (a+b\right )\left (a+b\right )=2\times\frac{1}{2}ab+\frac{1}{2}c^2$
Multiply both sides by $2$
$\left (a+b\right )^2=2ab+c^2$
$a^2+2ab+b^2=2ab+c^2$
$a^2+b^2=c^2$