The sides of the triangle are given as 1, x, and x².
The principle of triangle inequality requires that the sum of the lengths of any two sides should be equal to, or greater than the third side.
Consider 3 cases Case (a): x < 1, Then in decreasing size, the lengths are 1, x, and x². We require that x² + x ≥ 1 Solve x² + x - 1 = x = 0.5[-1 +/- √(1+4)] = 0.618 or -1.618. Reject the negative length. Therefore, the lengths are 0.382, 0.618 and 1.
Case (b): x = 1 This creates an equilateral triangle with equal sides The sides are 1, 1 and 1.
Case (c): x>1 In increasing order, the lengths are 1, x, and x². We require that x + 1 ≥ x² Solve x² - x - 1 = 0 x = 0.5[1 +/- √(1+4)] = 1.6118 or -0.618 Reject the negative answr. The lengths are 1, 1.618 and 2.618.
Answer: The possible lengths of the sides are (a) 0.382, 0.618 and 1 (b) 1, 1 and 1. (c) 2.618, 1.618 and 1.
