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PYTHAGOREAN TRIPLE It is surprising that there are some right-angled triangles where all three sides are whole numbers. The three whole number side-lengths are called a Pythagorean triple. An example is a = 3, b = 4 and c = 5, called "the 3-4-5 triangle". 32+42 = 9 + 16 = 25 = 52 so a2 + b2 = c2. Notice that the greatest common divisor of the three numbers 3, 4, and 5 is 1. Pythagorean triples with this property are called primitive. From primitive Pythagorean triples, you can get other, imprimitive ones, by multiplying each of a, b, and c by any positive whole number d > 1. This is because a2 + b2 = c2 if and only if (da)2 + (db)2 = (dc)2. Thus (a,b,c) is a Pythagorean triple if and only examples if (da,db,dc) is. For example, (6,8,10) and (9,12,15) are imprimitive Pythagorean triples.

Here are the of the first few primitive Pythagorean triples a b c a b c a b c 3 4 5 5 12 13 7 24 25 8 15 17 9 40 41 11 60 61 12 35 37 13 84 85 16 63 65 36 77 85 39 80 89 48 55 73

CHALLENGE YOURSELF

1. Find the only Pythagorean triangles with an area equal to their perimeter. 2. Ahmad has a triangular orchard with sides measuring 560 m, 420 m and 300 m. Determine whether the shape of his orchard is a right-angled, obtuse-angled or acute-angled triangle.