# Beyond Pythagoras

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Introduction

Beyond Pythagoras For this piece of coursework I am trying to find Pythagorean triplets (these are whole numbers that's satisfies Pythagoras theorem). Pythagoras Theorem is a2 + b2 = c2. (a) Being the shortest side, (b) being the middle side and (c) being the longest side (hypotenuse) of a right angled triangle. The numbers 3, 4 and 5 satisfy this condition 32 + 42 = 52 Because 32 = 3 x 3 = 9 42 = 4 x 4 = 16 52 = 5 x 5 = 25 = 32 + 42 = 9 + 16 = 25 = 52 To find the perimeter we add all the sides together. Perimeter = 3 + 4 + 5 = 12 Finally to find the areas we times the smallest and middle side and then divide by to. Area = 1/2 x 3 x 4 = 6 Odd Triples I will now put the first term in to a table and try to find the next terms up to 10. Term Number (n) Shortest Side (s) Middle Side (m) Longest Side (l) Perimeter (p) Area (a) 1 3 4 5 12 6 2 5 12 13 30 30 3 7 24 25 56 84 4 9 40 41 90 180 5 11 60 61 132 330 ...read more.

Middle

6n2 + 2n by 2 to get 2n3 + 3n2 + n = area (Also you can use the formula s x m divided by 2) Even Triples I have now moved on to find out right-angled triangles with an even positive integer for the shortest side. The smallest number, which I have found to be the smallest number, is 6. Below shows a table of all of the even triples I have found. shortest side (s) middle side (m) longest side (l) Perimeter (p) Area (a) 6 8 10 24 24 8 10 24 26 60 120 12 16 20 48 96 14 48 50 112 336 16 18 80 82 180 720 20 48 50 120 480 22 120 122 264 1320 24 26 168 170 364 2184 28 96 100 224 1344 30 224 226 480 3360 32 34 288 290 612 4896 36 160 164 360 2880 38 360 362 760 6840 40 . 42 440 442 924 9240 44 240 244 528 5280 When I was finding out even triples I could not find any for any number of a multiple of 8. I have noticed that some of the middle and longest sides have a difference of 2 and others 4. ...read more.

Conclusion

176 260 The difference between the original sequence and the formula is now 4 for each stage, so the final formula is: 8n� + 12n + 4 Area 24 120 336 720 1320 96 216 384 600 . 120 168 216 48 48 The difference of this sequence is 48 on line 3. This then means that you have to divide by three and then square the number, so you get 8n�. This formula is shown below. 8 64 216 512 1000 As this is not the same as the original sequence you will need to difference it again. This is shown below Difference = 16 56 120 208 320 40 64 88 112 24 24 24 This gives you 12n� because you have to half the end number then square it. This then gives you the formula 8n� + 12n�. The results of this formula are shown below. 20 112. 324 704 1300 The difference of the original sequence and the original formula are shown below. 4 8 12 16 20 4 4 4 4 : The difference then gives you 4n, which can then be added to finalise the formula, which is: 8n� + 12n + 4n Formula Shortest Side (s) 4n + 2! Middle Side (m) 4n� + 4n Longest Side (l) 4n� + 4n + 2 Perimeter (p) 8n� + 12n + 4 Area. (a) 8n� + 12n� + 4n ...read more.

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