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Geometry: Cyclic Quadrilateral and Secant

XAT 2011 · Quantitative Ability & Data Interpretation · Hard · Geometry

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In a city, there is a circular park. There are four points of entry into the park, namely - P, Q, R and S. Three paths were constructed which connected the points PQ, RS, and PS. The length of the path PQ is 10 units, and the length of the path RS is 7 units. Later, the municipal corporation extended the paths PQ and RS past Q and R respectively, and they meet at a point T on the main road outside the park. The path from Q to T measures 8 units, and it was found that the angle PTS is 60°. Find the area (in square units) enclosed by the paths PT, TS, and PS.

  1. A.

    36\sqrt{3}

  2. B.

    54\sqrt{3}

  3. C.

    72\sqrt{3}

  4. D.

    90\sqrt{3}

  5. E.

    None of the above

Answer

C

Explanation

By secant property, TQ⋅TP=TR⋅TS  ⟹  8×(10+8)=TR×(TR+7)  ⟹  144=TR2+7TR  ⟹  TR=9  ⟹  TS=16TQ \cdot TP = TR \cdot TS \implies 8 \times (10 + 8) = TR \times (TR + 7) \implies 144 = TR^2 + 7TR \implies TR = 9 \implies TS = 16. Area of ΔPTS=12×PT×TS×sin⁡(60∘)=12×18×16×32=723\Delta PTS = \frac{1}{2} \times PT \times TS \times \sin(60^\circ) = \frac{1}{2} \times 18 \times 16 \times \frac{\sqrt{3}}{2} = 72\sqrt{3}.

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