CAT 2020 Slot 1 QA Question & Solution

GeometryMedium

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

$\frac{6\pi}{25}$

$\frac{5\pi}{18}$

$\frac{3\pi}{25}$

$\frac{2\pi}{15}$

Let the length of radius be 'r'.

From the above diagram,

$x^2+r^2=6^2$....(i)

$(10-x)^2+r^2=8^2$----(ii)

Subtracting (i) from (ii), we get:

$x=3.6 \Rightarrow r^2=36-(3.6)^2 = 23.04$.

Area of circle = $\pi r^2 = 23.04\pi$

Area of rhombus $= \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 12 \times 16 = 96$.

$\therefore \text{ Ratio of areas} = 23.04\pi / 96 = \frac{6\pi}{25}$