# Pythagoras's Theorem/Proof 5

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## Theorem

Let $\triangle ABC$ be a right triangle with $c$ as the hypotenuse.

Then:

- $a^2 + b^2 = c^2$

## Proof

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

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

The one on the right has four triangles of area $\dfrac {a b} 2$ 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 $c^2 = a^2 + b^2$.

$\blacksquare$

## Source of Name

This entry was named for Pythagoras of Samos.

## Historical Note

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

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.1$: The Pythagorean Theorem - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Pythagoras' Theorem**

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