Points on Line Making Triangle

Points A, B, C, and D are distinct and lie, in given order, on a straight line. Line segments AB, AC, and AD have lengths x, y, and z, respectively. If line segments AB and CD may be rotated about points B and C, respectively, so that points A and D coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied?

  1. x < z/2
  2. y < x + z/2
  3. y < z/2

  4. Show/Hide Solution

    x + z – y > yx
    → 2x + z – 2y > 0
    → 2x + z > 2y
    x + z/2 > y, so statement II must be satisfied.

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