Pythagoras's Theorem/Proof 5

Theorem

Given any right triangle $\triangle ABC$ with $c$ as the hypotenuse, we have $a^2 + b^2 = c^2$.

Proof

The two squares both have the same area, that is, $\left({a + b}\right)^2$.

The one on the left has four triangles of area $\displaystyle \frac {ab} 2$ and a square of area $c^2$.

The one on the right has four triangles of area $\displaystyle \frac {ab} 2$ and two squares: one of area $a^2$ and one of area $b^2$.

Take away the triangles from both of the big squares and you are left with $c^2 = a^2 + b^2$.

$\blacksquare$

Historical Note

This proof is the basis of the Aldous Huxley short story Young Archimedes.

Sources

• George F. Simmons: Calculus Gems (1992), Chapter $\text {B}.1$
• For a video presentation of the contents of this page, visit the Khan Academy.