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28 September, 17:17

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

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  1. 28 September, 17:47
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    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.
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