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Area Ratio in Rectangle

XAT 2017 · Quantitative Ability & Data Interpretation · Hard · Geometry

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ABCD is a rectangle. P, Q and R are the midpoint of BC, CD and DA. The point S lies on the line QR in such a way that SR: QS = 1:3. The ratio of the area of triangle APS to area of rectangle ABCD is

  1. A.

    36/128

  2. B.

    39/128

  3. C.

    44/128

  4. D.

    48/128

  5. E.

    64/128

Answer

A

Explanation

Let AB=4,BC=2AB = 4, BC = 2. R=(0,1)R = (0, 1), Q=(2,0)Q = (2, 0), P=(4,1)P = (4, 1), A=(0,2)A = (0, 2), D=(0,0)D = (0, 0). SS divides RQRQ in ratio RS:SQ=1:3RS:SQ = 1:3. Coordinates of S=(3(0)+1(2)4,3(1)+1(0)4)=(0.5,0.75)S = \left(\frac{3(0) + 1(2)}{4}, \frac{3(1) + 1(0)}{4}\right) = (0.5, 0.75).

Area of △APS\triangle APS via coordinates: A=(0,2),P=(4,1),S=(0.5,0.75)A = (0, 2), P = (4, 1), S = (0.5, 0.75). Area =12∣0(1−0.75)+4(0.75−2)+0.5(2−1)∣=12∣−5+0.5∣=4.52=2.25=94= \frac{1}{2} | 0(1 - 0.75) + 4(0.75 - 2) + 0.5(2 - 1) | = \frac{1}{2} | -5 + 0.5 | = \frac{4.5}{2} = 2.25 = \frac{9}{4}.

Area of ABCD =4×2=8= 4 \times 2 = 8. Ratio =2.258=932=36128= \frac{2.25}{8} = \frac{9}{32} = \frac{36}{128}.

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