# Pythagorean Triangle/Example/3-4-5

## Example of Primitive Pythagorean Triangle

The triangle whose sides are of length $3$, $4$ and $5$ is a primitive Pythagorean triangle.

## Proof

 $\displaystyle 3^2 + 4^2$ $=$ $\displaystyle 9 + 16$ $\displaystyle$ $=$ $\displaystyle 25$ $\displaystyle$ $=$ $\displaystyle 5^2$

It follows by Pythagoras's Theorem that $3$, $4$ and $5$ form a Pythagorean triple.

Note that $3$ and $4$ are coprime.

Hence, by definition, $3$, $4$ and $5$ form a primitive Pythagorean triple.

The result follows by definition of a primitive Pythagorean triangle.

$\blacksquare$

## Historical Note

To the Pythagoreans, the $3-4-5$ triangle had particular significance: the sides of lengths $3$ and $4$ denoted the male and female principles, while the hypotenuse of lengths $5$ denoted their offspring.