# Pythagorean Triples

How to use the Pythagorean triples to solve the sides of a given right triangle: commonly used triples, examples, and their solutions.

## Triples

The Pythagorean triples are three positive integers
that satisfy the Pythagorean theorem:
a2 + b2 = c2.

Below triples are the commonly used triples.

(3, 4, 5): 32 + 42 = 52
(5, 12, 13): 52 + 122 = 132
(7, 24, 25): 72 + 242 = 252

## Example 1

Starting from the shortest side,
the sides of the given right triangle are (x, 4, 5).

Then the related triple is (3, 4, 5).

So x = 3.

## Example 2

Starting from the shortest side,
the sides of the given right triangle are (5, 12, x).

Then the related triple is (5, 12, 13).

So x = 13.

## Example 3

Starting from the shortest side,
the sides of the given right triangle are (6, x, 10).

Then the related triple is (3, 4, 5).

So draw a (3, 4, 5) right triangle
next to the given triangle.

These two triangles are similar.

Since these two triangles are similar,
their sides are proportional.

So x/4 = 6/3.

Similar triangles

6/3 = 2

Multiply 4 on both sides.

Then x = 8.