Pythagorean Triangle/Examples/6-8-10

Example of Pythagorean Triangle

The triangle whose sides are of length $6$, $8$ and $10$ is a Pythagorean triangle.

This is not a primitive Pythagorean triangle.

Proof

\(\ds 6^2 + 8^2\)

\(=\)

\(\ds 2^2 \times 3^2 + 2^2 \times 4^2\)

\(\ds \)

\(=\)

\(\ds 4 \times \paren {9 + 16}\)

\(\ds \)

\(=\)

\(\ds 4 \times 25\)

\(\ds \)

\(=\)

\(\ds 2^2 \times 5^2\)

\(\ds \)

\(=\)

\(\ds 10^2\)

It follows by Pythagoras's Theorem that $6$, $8$ and $10$ form a Pythagorean triple.

Note that $6$ and $8$ are not coprime as $\gcd \set {6, 8} = 2$.

Hence, by definition, $6$, $8$ and $10$ do not 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): $13$