If the lengths of the sides of a right angled triangle ABC right angled at C are in arithmetic progression, what is 5[sin(A)+sin(B)]?
Let a = BC = x-d, b = AC = x, and c = AB = x+d, where x > 0. Clearly d <> 0, for otherwise the triangle is equilateral. By the Pythagorean theorem,
c^2 = a^2+b^2,
(x+d)^2 = (x-d)^2+x^2,
(x+d)^2-(x-d)^2 = x^2,
4dx = x^2,
x(4d-x) = 0,
x = 4d.
Thus (a, b, c) = (3d, 4d, 5d). It follows that
5[sin(A)+sin(B)] = 5[3d/(5d)+4d/(5d)] = 7.
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