# The Pythagorean Theorem

One of the foremost fundamental results is that the well-known Pythagorean Theorem.

This states that a a2 + b2 = c2 in a right triangle with sides a and b and hypotenuse c.

The figure to the proper indicates one among the various known proofs of this fundamental result. Indeed, the area of the “big” square is (a + b)2 and can be decomposed into the area of the smaller square plus the areas of the four congruent triangles.

That is,

(a + b)2 = c2 + 2ab

which immediately reduces to a2 + b2 = c2

Next, we recall the equally well-known result that the sum of the interior angles of a triangle is 180 . The proof is easily inferred from the diagram to the right.

Exercises:

01.

Prove Euclid’s Theorem for Proportional Segments, i.e., given the right triangle ΔABC as indicated, then
h2 = pq, a2 = pc, b2 = qc

02. Prove that the sum of the interior angles of a quadrilateral ABCD is 360.

03.

In the diagram to the right, ΔABC is a right triangle, segments [AB] and [AF] are perpendicular and equal in length, and [EF] is perpendicular to [CE]. Set a = BC, b = AB, c = AB, and deduce President Garfield’s proof* of the Pythagorean theorem by computing the area of the trapezoid BCEF.

*James Abram Garfield (1831–1881) published this proof in 1876 in the Journal of Education (Volume 3 Issue 161). While a member of the House of Representatives. He was assassinated in 1881 by Charles Julius Guiteau. As an aside, notice that Garfield’s diagram also provides a simple proof of the fact that perpendicular lines in the planes have slopes which are negative reciprocals.

Related lessons: Trigonometric Identities