Pythagorean Triangle/Examples/5-12-13

Example of Primitive Pythagorean Triangle

The triangle whose sides are of length $5$, $12$ and $13$ is a primitive Pythagorean triangle.



Proof

\(\ds 5^2 + 12^2\) \(=\) \(\ds 25 + 144\)
\(\ds \) \(=\) \(\ds 169\)
\(\ds \) \(=\) \(\ds 13^2\)

It follows by Pythagoras's Theorem that $5$, $12$ and $13$ form a Pythagorean triple.


Note that $5$ and $12$ are coprime.

Hence, by definition, $5$, $12$ and $13$ form a primitive Pythagorean triple.

The result follows by definition of a primitive Pythagorean triangle.

$\blacksquare$


Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$
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