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Parallelogram side length constraint

CAT 2018 Slot 2 · Quantitative Ability · Easy · Geometry

This is an easy Quantitative Ability question from the CAT 2018 Slot 2 paper. It tests Geometry. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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A parallelogram ABCD has area 48 sq cm. If the length of CD is 8 cm and that of AD is ss cm, then which one of the following is necessarily true?

  1. A.

    s ≥ 6

  2. B.

    s ≠ 6

  3. C.

    5 ≤ s ≤ 7

  4. D.

    s ≤ 6

Answer

A

Explanation

Area of parallelogram =CD×AD×sin⁡θ=8×s×sin⁡θ= CD \times AD \times \sin\theta = 8 \times s \times \sin\theta, where θ\theta is the angle between sides CDCD and ADAD.

Given area =48= 48: 8ssin⁡θ=48  ⟹  ssin⁡θ=68s \sin\theta = 48 \implies s \sin\theta = 6 Since sin⁡θ≤1\sin\theta \le 1 for all θ\theta, we must have: s=6sin⁡θ≥6s = \frac{6}{\sin\theta} \ge 6

Therefore, s≥6s \ge 6 is necessarily true.

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