## Beyond Pythagoras

Introduction:

a² + b² = c²

The numbers 3, 4 and 5 can be the lengths of the sides of a right-angled triangle.

The perimeter = a + b + c The area = a × b ÷ 2

The numbers 5, 12 and 13 can also be the lengths of the sides of a right-angled triangle.

This is also true for 7, 24 and 25.

These numbers are all called Pythagorean triples because they satisfy the condition.

Aim: I am going to investigate the different values of a, b and c for which the formula a² + b² + c² works. I will also investigate the even and odd values for (a) for which the formula works. I also intend to find out a relationship between a, b and c and their perimeters and areas.

The rules for my first Pythagorean triples are:

- (a) Must be a odd consecutive number
- (b) Must be one smaller than c
- (c) Must be one bigger than b

I am now going to construct a table for these Pythagorean triples and I will try and work out patterns and formulas to help me work out the value of a, b and c and the perimeter and area of the triangles.

Formulas:

The formula for a is:

## N a

1 3 = 1 ×2 + 1

2 5 = 2× 2 + 1