Pythagoras' theorem

In a nutshell

Pythagoras' theorem (or the Pythagorean theorem) is a formula that involves 3 sides of a right-angled triangle.

The hypotenuse of a triangle is the longest side. For a right-angled triangle, the hypotenuse will be opposite the right angle.

Pythagoras' theorem states that

$a^2\,+b^2\,=c^2$

Where $c$ is the hypotenuse of the triangle, and $a$ and $b$ are the other 2 sides.

A right-angled triangle has the 2 shorter sides with length $5cm$ and $12cm$ . Find the length of the longest side.

Sketch the triangle:

Label the sides:

Use Pythagoras' theorem:

$a^2\,+b^2\,=c^2$

Replace $a$ and $b$ with your values, and solve:

$12^2+5^2=c^2$

$144+25=c^2$

$c^2=169$

$c=\sqrt{169}=13$

$\underline{c=13cm}$

Note: Don't forget to write the correct unit for your length!

A right-angled triangle has a hypotenuse of length $8cm$ , and a base of length $2cm$ . What is the area of the triangle to two decimal places?

First, work out the perpendicular height of the triangle using Pythagoras' theorem. Let $a=2$ and $c=8$ :

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

$2^2+b^2=8^2$ 

$b^2=8^2-2^2=64-4=60$ 

$b=\sqrt{60}cm$ 

Then, work out the area of the triangle using the area formula:

$\text{Area}=\dfrac{1}{2}\times\text{base}\times\text{perpendicular\,height}$ 

$\text{Area}=\dfrac{1}{2}\times2\times\sqrt{60}$ 

$\text{Area}=7.74596669...$ 

The area is $\underline{7.75 \ cm^2 \space (2\thinspace d.p)}$ .