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Semicircle Area with Perpendicular Chord

CAT 2006 Slot 1 · Quantitative Ability · Medium · Quantitative Aptitude

This is a medium Quantitative Ability question from the CAT 2006 Slot 1 paper. It tests Quantitative Aptitude. 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 semi-circle is drawn with ABAB as its diameter. From CC, a point on ABAB, a line perpendicular to ABAB is drawn meeting the circumference of the semi-circle at DD. Given that AC=2AC = 2 cm and CD=6CD = 6 cm, the area of the semi-circle (in sq. cm) will be:

  1. A.

    32 pi

  2. B.

    50 pi

  3. C.

    40.5 pi

  4. D.

    81 pi

  5. E.

    undeterminable

Answer

50 pi

Explanation

Since ∠ADB=90∘\angle ADB = 90^\circ (angle in a semicircle): In right triangle ADBADB, CD2=AC×CBCD^2 = AC \times CB. 62=2×CB  ⟹  36=2×CB  ⟹  CB=186^2 = 2 \times CB \implies 36 = 2 \times CB \implies CB = 18. Total diameter AB=AC+CB=2+18=20AB = AC + CB = 2 + 18 = 20 cm. Radius R=10R = 10 cm. Area of semi-circle =12πR2=12π(102)=50π= \frac{1}{2} \pi R^2 = \frac{1}{2} \pi (10^2) = 50\pi.

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