A circle is circumscribed around a square as shown in the figure. The area of one of the four shaded portions is equal to $$\frac{4}{7}$$. The radius of the circle is :
A. $$\sqrt 2 $$
B. $$\frac{1}{{\sqrt 2 }}$$
C. $$2$$
D. $$3$$
Answer: Option A
Solution(By Examveda Team)
Let the radius of the circle be r and the side of the square be aThen,
Diagonal of the square = 2r
⇒ $$\sqrt 2 $$ a = 2r
⇒ a = $$\frac{2}{{\sqrt 2 }}$$ r
⇒ a = $$\sqrt 2 $$ r
Area of one shaded portion :
$$\eqalign{ & = \frac{1}{4}\left[ {\pi {r^2} - {{\left( {\sqrt 2 r} \right)}^2}} \right] \cr & = \frac{1}{4}\left( {\pi - 2} \right){r^2} \cr & \therefore \frac{1}{4}\left( {\pi - 2} \right){r^2} = \frac{4}{7} \cr & \Rightarrow \left( {\frac{{22}}{7} - 2} \right){r^2} = \frac{{16}}{7} \cr & \Rightarrow {r^2} = \frac{{16}}{7} \times \frac{7}{8} \times 2 \cr & \Rightarrow r = \sqrt 2 \cr} $$
Related Questions on Area
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The percentage increase in the area of a rectangle, if each of its sides is increased by 20% is:
A. 40%
B. 42%
C. 44%
D. 46%
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