Home > Geometry, Triangles > Pythagoras’ Theorem

## Pythagoras’ Theorem

Definition: The square of the hypotenuse is equal to the sum of the squares of the other two sides.

#### Proof

What this means is that if we draw three squares where the length of the sides of the squares are equal to the lengths of the sides of the triangle the area of the big square equals the area of the two smaller squares added together.  (note: The hypotenuse is opposite the right angle and this side produces the largest square)

In the diagram above the largest square is the green square so this implies that $c^2=a^2+b^2$

It follows that if we are finding a shorter side then we could use $a^2=c^2-b^2$

Examples

Find the length of the hypotenuse.

Here $a=3\mbox{ and }b=7$

So using $c^2=a^2+b^2$

We get $c^2=3^2+7^2=58$

$\implies c=\sqrt{58}=7.62cm$

Find the length of the missing side

Since we are finding one of the shorter sides we will use $a^2=c^2-b^2$

Here $b=4\mbox{ and }c=12$

We get $a^2=12^2-4^2=128$

$\implies a=\sqrt{128}=11.31cm$

Find the value of $x$This is a right-angled triangle so let’s use $c^2=a^2+b^2$

$\implies 50^2=x^2+(2x)^2$

$\implies 2500=5x^2$

$\implies x^2=500$

$\implies x=\sqrt{500}=22.36mm$

Does 11cm, 59cm, 60cm produce a right-angled triangle?Use $c^2=a^2+b^2$ where $c=60$ since that is the longest length.

$\implies 60^2=11^2+59^2$

But $3600\neq 3602$, therefore this is not a right-angled triangle.