## Pythagoras' Theorem Pythagoras was a teacher in Greece about 2 500 years ago. He discovered a connection between the longest side of a right-angled triangle (called the hypotenuse ) and the other two sides.

He noticed that the area of a square on the hypotenuse was equal to the sum of the areas of the squares on the two other sides. Another way to show this connection is by measuring the lengths of the sides of two different triangles

 Triangle 1 Triangle 2  Length of AB = 3.8 cm (a short side) Length of AB = 6.6 cm (a short side) Length of BC = 3.2 cm (a short side) Length of BC = 1.9 cm (a short side) (Length of AB)2 + (Length of BC)2 = 14.44 + 10.24 = 24.68 (Length of AB)2 + (Length of BC)2 = 43.56 + 3.61 = 47.17 Length of AC = 4.9 cm (the longest side) (Length of AC)2 = 24.01 Length of AC = 6.9 cm (the longest side) (Length of AC)2= 47.61 The results show that the two short sides, squared and added are the same (almost) as the longest side squared. (The small difference in the two results could be due to only being able to measure to the nearest millimetre or inaccurate measuring.)

### Pythagoras Theorem

Pythagoras' findings are called the Theorem of Pythagoras.

It states that in a right-angled triangle the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the two other sides (the two shorter sides). h2 = a2 + b2

Note: h, the hypotenuse is always opposite the right angle.

The converse of the theorem is also true

i.e. If h2 = a2 + b2 , then the triangle is right-angled.

 Examples Answers (a) Find the length of the side marked h  (b) Find the length of the side marked x  Pythagorean Triples

A Pythagorean triple is a set of numbers that satisfy the Pythagoras equation, given above.

Some common ones are {3, 4, 5}, {5, 12,13} and {7, 24, 25}. These should be memorised.

Multiples of these triples also obey Pythagoras' Theorem.

e.g. {6, 8, 10} and {9, 12, 15}

Example

Are 7, 24 and 26 a Pythagorean triple?

If they are then 72 + 242 should be the same as 262

72 + 242 = 676

262 = 676

Therefore, 7 , 24 and 26 are a Pythagorean triple.