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I was set the task of investigating families of Pythagorean triples. I started off by investigating families of Pythagorean triples where the shortest side is an odd number and c is always equal to b+1. I then extended my research into other families of triples. Finally, I came up with a formula that can calculate all Pythagorean triples. Pythagoras's formula is as follows: a2 + b2 = c2 This is referring to such a triangle: The triangle always has one right angle and c is always the longest side, also known as the hypotenuse. An example of this is with the triple 3,4 and 5: Here, a is 3, b is 4 and c is 5. 52 = 32 + 42 25 = 9 + 16 I will now investigate the Pythagorean triples where c is always equal to b+1. The following table shows the first ten cases, along with the areas and perimeters: N a b c P A a2 b2 c2 1 3 4 5 12 6 9 16 25 2 5 12 13 30 30 25 144 169 3 7 24 25 56 84 49 576 625 4 9 40 41 90 180 81 1600 1681 5 11 60 61 132 330 121 3600 3721 6 13 84 85 182 546 169 7056 7225 7 15 112 113 240 840 225 12544 12769 8 17 144 145 306 1224 289 20736 21025 9 19 180 181 380 1710 361 32400 32761 10 21 220 221 462 2310 441 48400 48841 By looking at the...

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