# Beyond Pythagoras

## 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:

1. (a) Must be a odd consecutive number
2. (b) Must be one smaller than c
3. (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

3                       7          = 3 × 2 + 1

4                       9          = 4 ×2 + 1

5                      11               = 5 × 2 + 1

n × 2 + 1

So from this I can work out that the general rule for A is 2n + 1.

Now I will try and work out a formula for B.

So the formula for B is n² + n × 2 + n²

C is one bigger than ...