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UPSC CDS 1 2023 Math Paper

Question Numbers: 97-98

Consider the following for the next two (02) items that follow :

In the following figure, a rectangle ABCD is inscribed in a circle of radius

r

. Given

\angle DAE = 30^{\circ}

and

\angle ACD = 30^{\circ}

Question : 97 of 100

Marks: +1, -0

What is the ratio of the area of the circle to the area of the rectangle ?

Solution:

Concept:

In a

30^{\circ}-60^{\circ}-90^{\circ}

right triangle, the sides opposite these angles are in the ratio

1 : \sqrt{3} : 2

Explanation:

Given the arrangement, triangle

\text{ADC}

is right-angled at

D

with angles

30^{\circ}

60^{\circ}

90^{\circ}

Let the circle's radius be

r

From the side ratio, we have

\text{AD} : \text{DC} : \text{AC} = r : r\sqrt{3} : 2r

Thus the rectangle has sides

\text{AD} = r

and

\text{DC} = r\sqrt{3}

Area of circle =

\pi r^2

Area of rectangle =

\text{AD} \times \text{DC} = r \times r\sqrt{3} = r^2\sqrt{3}

Required ratio =

\frac{\pi r^2}{r^2\sqrt{3}} = \frac{\pi}{\sqrt{3}}

Answer:

\frac{\pi}{\sqrt{3}}

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