• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Beyond Pythagoras

Extracts from this document...

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.

The above preview is unformatted text

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Pythagorean Triples essays

  1. Maths Number Patterns Investigation

    I can now safely say that 4n2 - 4(n-1)2 is definitely not the correct formula for the middle side. I believe the problem with 4n2 - 4(n-1)2 was that 4n2, once you start using larger numbers, becomes far to high to bring it back down to the number that I want for the middle side.

  2. Pythagoras Theorem

    2a� + 1 = a4 + 2a� + 1 = a4 + 2a� + 1 = a4 +2a� +1 By cancelling the formula down, so that it is the same on both sides, I have proven that the formulae: o b = (a� - 1)/2 o c = (a� +

  1. Beyond Pythagoras

    61 6 13 84 85 7 15 112 113 8 17 144 145 9 19 180 181 10 21 220 221 11 23 264 265 12 25 312 313 13 27 364 365 14 29 420 421 15 31 480 481 16 33 544 545 17 35 612 613 18

  2. Investigate the area of triangle studies including the Pythagorean Theorem and in particular Pythagorean ...

    However, because 4 are the difference of the difference, the formula must be n�. I now believe that the answer will have something to do with 4n�So, I will now write out the answers for 4n� 4n�works for the first term, but, it then collapses after this, as the difference

  1. Beyond Pythagoras.

    N Length of middle side 1 4 2 12 3 24 There does seem to be a pattern because all the middle side numbers are multiples of 4. 12-4=8 24-12=12 12-8=4. This again reinforces the fact that there is a pattern as the 2nd difference is 4.

  2. Investigating families of Pythagorean triples.

    400 2 20 48 52 400 2304 2704 3 28 96 100 784 9216 10000 4 36 160 164 1296 25600 26896 5 44 240 244 1936 57600 59536 6 52 336 340 2704 112896 115600 7 60 448 452 3600 200704 204304 8 68 576 580 4624 331776 336400

  1. For this piece of work I am investigating Pythagoras. Pythagoras was a Greek mathematician.

    This must mean that the formula is: 2n�+2n I think this is the formula from looking at the first few terms but I will check it using another random number. The number will be 18. 2x18�+2x18=684 This is the same as the number given in the table so I presume that it is right.

  2. Beyond Pythagoras

    n a Difference 1 1 3 2 1 2 5 2 1 3 7 2 1 4 9 2 The differences between the numbers show that `a' is changing twice as fast as `n'. Therefore, 2n must be part of the formula.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work