UPSC CDS 2 2019 Math Paper

Let ABCD be the square inscribed in the semi-circle with center 0 , and side CD the diameter of the semi-circle.
Let the point M be the mid-point of AB. Now, join OB and OM to get the right △OMB with OB as the hypotenuse. Therefore,
OM2+MB2=OB2
OM= side of the square inscribed in the semi-circle, MB= half of the side; and
OB= radius
Let the side of the square be x.
x2+(‌

x

2

)2=r2
x2=‌

4r2

5

Hence, the area of the semi-circle =‌

4r2

5

Now diagonal of the square inscribed in the circle =2r
Therefore, its area =‌

(2r)2

2

=2r2
Area of the square =‌

‌ diagonal ‌2

2

Hence, the required ratio =‌

4r2

5

:2r2=‌

2

5

:1=2:5