President Garfield

1)         President Garfield’s Proof of the Pythagorean Theorem

References: 

Loomis, Elisha S. (1968). The Pythagorean proposition. Washington, DC: National Council of Teachers of Mathematics.

Multiplication By Infiniti. (2011). US president garfield’s proof of the Pythagorean theorem. Retrieved from http://tetrahedral.blogspot.com/2011/04/us-president-garfields-proof-of.html

In the figure shown below, we have taken an arbitrary right triangle with sides of length a, b, and hypotenuse of length c. We draw a second copy of this same triangle ABC as pictured below.

 

First, we need to find the area of the trapezoid by using the area formula of the trapezoid.

Formula for the area of a Trapeziod  A = ½ h(b₁ + b₂)

Given the above diagram, h= a + b, b₁=a, and b₂=b.

Using substitution and the area formula for a trapezoid:

A=½(a+b)(a+b)
A =½(a²+2ab+b²).

Now, let's find the area of the trapezoid by summing the area of the three right triangles.

The area of the yellow triangle:
A_y = ½ ab

The area of the white triangle:
A_w = ½c²

The area of the blue triangle:
A_b = ½ ab

We find the sum of the three triangles:
½ ab + ½c²+ ½ ab = ½ (ab + c² + ab) = ½ (2ab + c²).

Since the area of the trapezoid equals the sum of the areas of the three triangles, we can set up the following equality:

½(a² + 2ab + b²) = ½ (2ab + c²).

Multiplying both sides by 2:

a² + 2ab + b² = 2ab + c²

And, subtracting 2ab from both sides:

a² + b² = c²

Thus, we are able to prove the Pythagorean Theorem.