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Trapezoid Side Ratios and Sine Function

XAT 2015 · Quantitative Ability & Data Interpretation · Hard · Quantitative Ability

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The parallel sides of a trapezoid ABCD are in the ratio of 4:54 : 5. ABCD is divided into an isosceles triangle ABP and a parallelogram PBCD. ABCD has a perimeter equal to 1120 meters1120\text{ meters} and PBCD has a perimeter equal to 1000 meters1000\text{ meters}. Find sin⁡∠ABC\sin \angle ABC, given 2∠DAB=∠BCD2\angle DAB = \angle BCD.

  1. A.

    4/5

  2. B.

    16/25

  3. C.

    5/6

  4. D.

    24/25

  5. E.

    A single solution is not possible

Answer

A

Explanation

Using perimeter relations: AP=AB=120AP = AB = 120, BP=100BP = 100. In △ABP\triangle ABP, applying cosine rule for ∠A\angle A gives cos⁡A=3/5  ⟹  sin⁡A=4/5\cos A = 3/5 \implies \sin A = 4/5. Since ∠ABC=180∘−∠A\angle ABC = 180^\circ - \angle A, sin⁡∠ABC=sin⁡A=4/5\sin \angle ABC = \sin A = 4/5.

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