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Introduction

## 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.

Middle

112

113

240

840

8

17

144

145

306

1224

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.

 B 4 12 24 40 N 1 2 3 4 N² 1 4 9 16 Difference between b and n 2 4 6 8

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

C is one bigger than B so the formula for C is n²  + n × 2  + n²  + 1.

The formula for P = a + b + c so that is (2n + 1)+(n²  + n × 2 + n² )+(n²  + n × 2 +n² + 1) = P

For example 7 + 24 + 25  = 56

The formula for area is A × B  ÷  2 so that means (2n + 1)(n²   +  n ×

Conclusion

 B 8 15 24 35 N 1 2 3 4 N² 1 4 9 16 Difference between b and n 3 8 15 19

So the formula for B is n²  + 4n + 3

C is two bigger than B so the formula for C is n² + 4n + 3  + 2.

The formula for P = a + b  + c so that is (2n + 4) + (n² + 4n + 3 ) + (n² + 4n + 3  + 2) = P .

For example 6 + 8 + 10 =  24

The formula for area is A ×  B ÷ 2 so that means (2n + 4)(n²  +  4n + 3) ÷ 2.

For example 6 × 8 ÷ 2 = 24

Part 3:

I will now change the rules again:

1. (a) Must be a multiple of 3
2. (b) Must be a multiple of 4
3. (c) Must be a multiple of 5

I will also try out my formulas from my previous Pythagorean triples and see if they work.

 N A B C P Area 1 3 4 5 12 6 2 6 8 10 24 24 3 9 12 15 36 54 4 12 16 20 48 96 5 15 20 25 60 150 Hi h

This student written piece of work is one of many that can be found in our GCSE Pythagorean Triples section.

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