2021 · Q49
A pole on the ground leans at 60° with the vertical. At a point x metre away from the base of the pole on the ground, two halves of the pole subtend the same angle. If the pole and the point are in the same vertical plane, then what is the length of the pole?
Elementary MathematicsTrigonometry
- A√2x metre
- B√3x metre
- C2x metre
- D2√2x metre
Explanation
Let the pole have length L, leaning 60° from the vertical (30° from the ground), and let X be x metres from the base on the leaning side. The equal angles mean XM bisects ∠BXT; since it meets BT at its midpoint M, the angle-bisector theorem gives BX = XT. So x² = (L√3/2 − x)² + (L/2)², which simplifies to L² = √3·Lx, i.e. L = √3·x. Hence option B.