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CAT 2020 Slot 1 — QA Question 22

Basics of QuadrilateralsEasy

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

Answer & solution

  • A

    5π/18

  • B

    2π/15

  • C

    3π/25

  • 6π/25

Solution

Easy

The rhombus area is half the product of diagonals. The inscribed circle's radius equals the distance from the centre to a side — which is the altitude from the right angle of the triangle formed by half-diagonals (legs 66 and 88, hypotenuse == side 1010). Then take the ratio of the two areas.

8 6
1

Area of the rhombus. Half the product of the diagonals.

Arhombus=12×12×16=96\begin{aligned} &A_{\text{rhombus}} = \tfrac12\times 12\times 16 = 96 \end{aligned}
2

Side and radius. Half-diagonals 66 and 88 form a right triangle; the side is its hypotenuse, and the in-radius equals the altitude from the right angle.

side=62+82=10r=6×810=245(altitude =leg⋅leghyp)\begin{aligned} &\text{side} = \sqrt{6^2+8^2} = 10\\ &r = \frac{6\times 8}{10} = \frac{24}{5} \quad\text{(altitude } = \tfrac{\text{leg}\cdot\text{leg}}{\text{hyp}}\text{)} \end{aligned}
3

Area of the circle and the ratio.

Acircle=πr2=π(245)2=576π25AcircleArhombus=576π/2596=576π2400=6π25\begin{aligned} &A_{\text{circle}} = \pi r^2 = \pi\left(\frac{24}{5}\right)^2 = \frac{576\pi}{25}\\ &\frac{A_{\text{circle}}}{A_{\text{rhombus}}} = \frac{576\pi/25}{96} = \frac{576\pi}{2400} = \frac{6\pi}{25} \end{aligned}
AcircleArhombus=6π25— option (d)\frac{A_{\text{circle}}}{A_{\text{rhombus}}} = \frac{6\pi}{25}\quad\text{— option (d)}
CAT 2020 Slot 1 QA Q22: A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area — Solution | TheCATExam