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Beyond Pythagoras

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Introduction

Beyond Pythagoras Pythagoras Theorem is a� + b� = c�. 'a' being the shortest side, 'b' being the middle side and 'c' being the longest side (which is always the hypotenuse) of a right angled triangle. The numbers 3, 4 and 5 satisfy this condition: 3� + 4� = 5� because 3� = 3 x 3 = 9 4� = 4 x 4 = 16 5� = 5 x 5 = 25 and so 3� + 4� = 9 + 16 = 25 = 5� We also checked to see if similar sets of numbers also satisfy this condition: (smallest number)� + (middle number)� = (largest number)� The numbers 5, 12 and 13 also satisfy this condition: 5� + 12� = 13� because 5� = 5 x 5 = 25 12� = 12 x 12 = 144 13� = 13 x 13 = 169 and so 5� + 12� = 25 + 144 = 169 = 13� The numbers 7, 24 and 25 also satisfy this condition: 7� + 24� = 25� because 7� = 7 x 7 = 49 24� = 24 x 24 = 576 25� ...read more.

Middle

n To get these formulas I did the following: Take side 'a' for the first five sets of numbers; 3, 5, 7, 9, 11. From these numbers you can see that the formula is 2n + 1 because they are consecutive odd numbers. From looking at my table of results, I noticed that 'an + n = b'. So I took my formula for 'a' (2n + 1) multiplied it by 'n' to get '2n� + n'. I then added my other 'n' to get: 2n� + 2n. Side 'c' is just the formula for side 'b' +1: 2n� + 2n + 1 The perimeter = a + b + c. Therefore I took my formula for 'a' (2n + 1), my formula for 'b' (2n� + 2n) and my formula for 'c' (2n� + 2n + 1). Then I did the following: 2n + 1 + 2n� + 2n + 2n� + 2n + 1 This can be rearranged to equal: 4n2 + 6n + 2 The area = (a x b) ...read more.

Conclusion

= (a + 2d)� a� + a� + ad + ad + d� = (a + 2d)� 2a� + 2ad + d� = (a + 2d)� 2a� + 2ad + d� = (a + 2d)(a + 2d) 2a� + 2ad + d� = a� + 2ad + 2ad + 4d� 2a� + 2ad + d� = 4d� + a� + 4ad If you equate these equations to 0 you get the following: a� - 3d� - 2ad = 0 Change a to x: x� - 3d� - 2dx = 0 Factorise this equation to get: (x + d)(x - 3d) Therefore: x = -d x = 3d x = -d is impossible as you cannot have a negative dimension. a, a+d, a + 2d Is the same as: 3d, 4d, 5d This tells us that the only Pythagorean triples are 3, 4, 5 or multiples of 3, 4, 5 e.g. 6, 8, 10 or 12, 16, 20 etc. Mathematics GCSE Coursework Beyond Pythagoras Luke Hopwood 11B Candidate number: 7484 The Mirfield Free Grammar ...read more.

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