# Pythagorean Triple

How to find the side of a right triangle by using the Pythagorean triple: definition, 3 examples, and their solutions.

## Definition

### Definition

The Pythagorean triple is 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

### Example

### Solution

The sides of the given right triangle are

(x, 4, 5).

The related triple is

(3, 4, 5).

So x = 3.

So x = 3 is the answer.

## Example 2

### Example

### Solution

The sides of the given right triangle are

(5, 12, x).

The related triple is

(5, 12, 13).

So x = 13.

So x = 13 is the answer.

## Example 3

### Example

### Solution

The sides of the given right triangle are

(6, x, 10).

(6, x, 10) looks like the multiple of (3, 4, 5).

So draw a (3, 4, 5) right triangle.

These two triangles are similar.

Then their sides are proportional.

So x/4 = 6/3.

Similar Triangles

6/3 = 2

x/4 = 2

Multiply 4 to both sides.

Then x = 8.

So x = 8.