... units ? The numbers 5,12,13 can also be the lengths - in appropriate units - of a right-angled triangle. Perimeter = 5+12+13=30 Area=?x5x12=30 This is also true for the numbers 7,24,25 Perimeter = 7+24+25=56 Area=?x7x24=84 I have put these results into a table to see if I can work out any patterns. Length of shortest side Length of middle side Length of longest side Perimeter Area 3 4 5 12 6 5 12 13 30 30 7 24 25 56 84 I have noticed that there seems to be a recurring pattern between the length of the middle side and the length of the longest side. They are always consecutive numbers. I shall now investigate this. I will assume that the hypotenuse has length b + 1 where b is the length of the middle side. (c=b+1) b+1 a b Using Pythagoras: a? + b? = (b+1) ? Expanding this: a? + b? = b? + 2b+ 1 therefore a? = 2b +1 This means that a? must be odd because it equals 2b + 1. Since a? is odd this means that a must be odd also. (even x even ...

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