# Pythagorean Triangle/Example/3-4-5

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## 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.

$\blacksquare$

## Also see

- Smallest Pythagorean Triangle is 3-4-5
- Pythagorean Triangle whose Area is Half Perimeter: its area and semiperimeter are both $6$
- Pythagorean Triangle with Sides in Arithmetic Progression

## 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.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $6$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $6$