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Ratio of area of inscribed circle to rhombus

CAT 2020 Slot 1 · Quantitative Ability · Hard · Geometry

This is a hard Quantitative Ability question from the CAT 2020 Slot 1 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 circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of the circle to the area of the rhombus is

  1. A.

    \frac{2\pi}{15}

  2. B.

    \frac{6\pi}{25}

  3. C.

    \frac{3\pi}{25}

  4. D.

    \frac{5\pi}{18}

Answer

B

Explanation

Diagonals of the rhombus d1=12d_1 = 12 cm and d2=16d_2 = 16 cm.

  1. Area of rhombus: Area=12d1d2=12×12×16=96 cm2\text{Area} = \frac{1}{2} d_1 d_2 = \frac{1}{2} \times 12 \times 16 = 96\text{ cm}^2

  2. Side of rhombus (ss): s=(d12)2+(d22)2=62+82=10 cms = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} = \sqrt{6^2 + 8^2} = 10\text{ cm}

  3. Inradius of circle (rr): Area=r×semi-perimeter  ⟹  96=r×(4×102)=20r\text{Area} = r \times \text{semi-perimeter} \implies 96 = r \times \left(\frac{4 \times 10}{2}\right) = 20r r=9620=245 cmr = \frac{96}{20} = \frac{24}{5}\text{ cm}

  4. Area of circle: Areacircle=πr2=π(245)2=576π25\text{Area}_{\text{circle}} = \pi r^2 = \pi \left(\frac{24}{5}\right)^2 = \frac{576\pi}{25}

  5. Ratio: Ratio=576π/2596=6π25\text{Ratio} = \frac{576\pi / 25}{96} = \frac{6\pi}{25}

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