Pythagorean Theorem Proof by an American President!

The Pythagorean Theorem has several different proofs. Here is one interesting & an entirely different proof by the $20^{th}$ President of the United States, James Garfield. He used three right triangles (ABE, AED, DEC) to make a trapezium.

Area of the trapezium

[ABCD]

\frac {1}{2}

(sum of the lengths of parallel sides)

\times

height

= \frac {1}{2} (p+q) (p+q) = \frac {p^2+q^2+2pq}{2}

Now, sum of the areas of the three triangles = $\frac{pq}{2}+\frac{pq}{2}+\frac{r^2}{2}$

Therefore, $\frac{pq}{2}+\frac{pq}{2}+\frac{r^2}{2}$ $= \frac {p^2+q^2+2pq}{2}$

$\Rightarrow 2pq+r^2 = p^2+q^2+2pq$

$\Rightarrow r^2 = p^2+q^2$

#PythagoreanTheorem #Interesting

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$