Pythagorean Triangle/Example/3-4-5

From ProofWiki
Jump to navigation Jump to search

Example of Primitive Pythagorean Triangle

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


3-4-5.png


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


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