Pythagoras’ Theorem

Pythagoras was born at Samos, in Greece, and lived from 580 to 500 B.C. He was a Mathematician who became famous for discovering something unique about right-angled triangles.

Let’s have a look at what he discovered…

So what is Pythagoras’ Theorem?

Well, he said that…

*“For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.”*

Consider squares being drawn on each side of a right-angled triangle…

See diagram below.

The formula for finding the longest side on a right-angled triangle is – **a²**** **** + b²**** ****= c²**** **

The longest side I called the ‘Hypotenuse’ and is always the side opposite the right-angle.

Below is an example.

Find the length of the hypotenuse

- Add 3² = 9 and 4² = 16

- The sum of the two numbers is 25
- Finally, square root
*x*=**√25**

*x ***= 5cm**

You can also find one of the shorter sides of a right-angled triangle by rearranging the formula.

The formula to find the shorter side is – **a²**** ****= c²**** ****– b²**

The shorter sides are always adjacent to the right angle.

Below is an example.

Find the length of the side marked x

- Subtract 3² = 9 from 5² = 25
- The answer is
*x²*= 16 - Square root
*x*=**√16**

*x ***= 4 cm **

**Remember…**

Pythagoras’ theorem can only be used in right-angled triangles.

Pythagoras’ theorem can be used to solve certain real-life practical problems, when a problem involves two lengths.

Below is a real-life Pythagoras problem.

A plane leaves Manchester airport and heads due east. It files 160 km before turning due north. It then files a further 280 km and lands. What is the distance of the return flight if the plane flies straight back to Manchester airport?

First, sketch the situation.

Using Pythagoras’ theorem gives:

*X²* = 1602 + 2802 km

= 25600 + 78400 km

= 104000 km²

So x = √104000 = 322 km

Pythagoras’ theorem is interesting in its own right but is very important in other areas.

Here’s some Pythagoras trivia….

**Trivia 1:**

Pythagoras’ theorem has appeared in an episode of *The Simpsons*, when it was quoted incorrectly by Homer.

**Trivia 2:**

In the year 2000, Uganda issued a 2000 shilling coin in the shape of a right-angled triangle. It had an image of Pythagoras and a statement of his theorem on one side.

**Trivia 3:**

In 1940 Elisha Scott Loomis published a book containing 256 different proofs of Pythagoras’ theorem. One of them was written by James Garfield, president of the United States, in 1881.