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Ratio of Heights on Vertical Pole

XAT 2021 · Quantitative Ability & Data Interpretation · Medium · Trigonometry

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The topmost point of a perfectly vertical pole is marked A. The pole stands on a flat ground at point D. The points B and C are somewhere between A and D on the pole. From a point E, located on the ground at a certain distance from D, the points A, B and C are at angles of 60, 45 and 30 degrees respectively. What is AB : BC : CD?

  1. A.

    (3+3):(1+3):1(3 + \sqrt{3}) : (1 + \sqrt{3}) : 1

  2. B.

    (3−3):1:(3−1)(3 - \sqrt{3}) : 1 : (\sqrt{3} - 1)

  3. C.

    1 : 1 : 1

  4. D.

    (3−3):(3−1):1(3 - \sqrt{3}) : (\sqrt{3} - 1) : 1

  5. E.

    (3−1):1:(3−3)(\sqrt{3} - 1) : 1 : (3 - \sqrt{3})

Answer

D

Explanation

Let ED=x3ED = x\sqrt{3}. In △CDE\triangle CDE: tan⁡30∘=CDED  ⟹  CD=x3tan⁡30∘=x\tan 30^\circ = \frac{CD}{ED} \implies CD = x\sqrt{3} \tan 30^\circ = x. In △BDE\triangle BDE: tan⁡45∘=BDED  ⟹  BD=x3\tan 45^\circ = \frac{BD}{ED} \implies BD = x\sqrt{3}. In △ADE\triangle ADE: tan⁡60∘=ADED  ⟹  AD=3x\tan 60^\circ = \frac{AD}{ED} \implies AD = 3x. BC=BD−CD=x3−x=x(3−1)BC = BD - CD = x\sqrt{3} - x = x(\sqrt{3}-1). AB=AD−BD=3x−x3=x(3−3)AB = AD - BD = 3x - x\sqrt{3} = x(3-\sqrt{3}). Ratio AB:BC:CD=(3−3):(3−1):1AB : BC : CD = (3-\sqrt{3}) : (\sqrt{3}-1) : 1.

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