**The Pythagorean Theorem (Proof):**

James Garfield (1876), later president of the United States, while serving in the House of Representatives, discovered the proof of the Pythagorean Theorem, using trapezoid.

Suppose, trapezoid bases are ‘a’ and ‘b’ and height (altitude) is equal to ‘(a+b)’.

Therefore,

Area of the trapezoid =

In addition, as shown in figure, trapezoid consists of three triangles 1, 2, and 3.

Combined Area of Triangles 1, 2, and 3 =

Since, combined area of triangles is equal to trapezoid area, therefore,

**A real-world application problem: **

A real world application example of the Pythagorean Theorem is designing of a ramp for disable persons.

**Question:** Assume that ramp height is 15 feet and suggested ratio (i.e. slope) for designing of ramp for disable person is approximately 1/12 (given). Calculate ramp length.

The ramp length can be calculated as follows:

Length of base of ramp = 15*12 = 180 feet.

Using Pythagorean Theorem,

Length of ramp = = =≈ 180.624 feet

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**Reference:**

Weisstein, Eric W. “Pythagorean Theorem.” From *MathWorld*–A Wolfram Web Resource. http://mathworld.wolfram.com/PythagoreanTheorem.html