CAT 2020 Quant Question [Slot 1] with Solution 21

Question

A circle is inscribed in a Rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is

  1. \(\frac{{5\pi }}{{18}}\)
  2. \(\frac{{6\pi }}{{25}}\)
  3. \(\frac{{3\pi }}{{25}}\)
  4. \(\frac{{2\pi }}{{15}}\)
Option: 2
Solution:

Given the circle is inscribed in the rhombus of diagonals 12 and 16 . Let O be the point of intersection of the diagonals of the rhombus. Then, OE (radius) \( \bot \) DC.

Also \(DC = \sqrt {{6^2} + {8^2}}  = 10\)

As area of \(\Delta ODC\) should be the same, we have, \(\frac{1}{2} \times 6 \times 8 = \frac{1}{2} \times OE \times 10\)

\( \Rightarrow OE = 4.8\)

\(\therefore \) Required ratio of areas \( = \frac{{\pi {{(4.8)}^2}}}{{\frac{1}{2} \times 12 \times 16}} = \frac{{6\pi }}{{25}}\)

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