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Geometry: Maximum Area of Quadrilateral

XAT 2016 · Quantitative Ability & Data Interpretation · Hard · Quantitative Ability

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ABCD is a quadrilateral such that AD = 9 cm, BC = 13 cm and ∠DAB=∠BCD=90∘\angle DAB = \angle BCD = 90^\circ. P and Q are two points on AB and CD respectively, such that DQ : BP = 1 : 2 and DQ is an integer. How many values can DQ take, for which the maximum possible area of the quadrilateral PBQD is 150 sq.cm?

  1. A.

    14

  2. B.

    12

  3. C.

    10

  4. D.

    9

  5. E.

    8

Answer

D

Explanation

Let DQ=xDQ = x, then BP=2xBP = 2x. Area of △BPD=12⋅AD⋅BP=12⋅9⋅2x=9x\triangle BPD = \frac{1}{2} \cdot AD \cdot BP = \frac{1}{2} \cdot 9 \cdot 2x = 9x. Area of △BQD=12⋅BC⋅DQ=12⋅13⋅x=6.5x\triangle BQD = \frac{1}{2} \cdot BC \cdot DQ = \frac{1}{2} \cdot 13 \cdot x = 6.5x. Area of PBQD=9x+6.5x=15.5x≤150PBQD = 9x + 6.5x = 15.5x \le 150. 15.5x≤150  ⟹  x≤15015.5≈9.6715.5x \le 150 \implies x \le \frac{150}{15.5} \approx 9.67 Since DQ=xDQ = x must be a positive integer, xx can take values 1,2,3,…,91, 2, 3, \dots, 9 (9 values).

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