The sides of a triangle are 1, x, and x2. what are possible values of x?

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.