# 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:*a*^{2} + *b*^{2} = *c*^{2}.

Below triples are the commonly used triples.

(3, 4, 5): 3^{2} + 4^{2} = 5^{2}

(5, 12, 13): 5^{2} + 12^{2} = 13^{2}

(7, 24, 25): 7^{2} + 24^{2} = 25^{2}

## 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.