Pythagorean Triples and Similar Triangles

What is a Pythagorean triple?

To best understand Pythagorean triples, we had best talk briefly about the Pythagorean Theorem first. The Pythagorean Theorem says that if a right triangle has a hypotenuse of side length c, and legs of lengths a and b, then the following formula must be true:

Now, a Pythagorean triple is a set of three natural numbers (a, b, c) (called an ordered triple) such that they satisfy the Pythagorean Theorem. The order in which the numbers are listed matter. If a, b, and c are plugged into the above equation, switching which number equals a, b, and c will ruin the equality.

There are lots of examples of Pythagorean Triples. Perhaps at this point you’d want to pause and try to find some. If you wish to try this, pause before you keep reading. My hint is to start by trying small numbers, and maybe have a calculator on hand.

Okay, so there are a lot of examples of Pythagorean triples, but here are a few:

(3, 4, 5), (6, 8, 10), (5, 12, 13), (8, 15, 17), (11, 60, 61)

As we move along I’ll talk about a place where some plane geometry helps us generate a lot of Pythagorean triples.

What are similar triangles?

Switching gears to more geometric definitions for a moment, one way to compare two triangles is to see if they are similar. Two triangles are similar if their side lengths are in direct proportion to each other. Let ABC and DEF be triangles…