The figure shows the rectangle ABCD with a semicircle and a circle inscribed inside in it as shown. What is the ratio of the area of the circle to that of the semicircle?
Explanation:
Let radius of the semicircle be R and radius of the circle be r. Let P be the centre of semicircle and Q be the centre of the circle. Draw QS parallel to BC. Now, ΔPQS ~ ΔPBC
∴PQPB=QSBC⇒R+r2R=R-rR⇒R+r=2R-2r⇒r(1+2)=R(2-1)
⇒r=R(2-1)(2+1)×(2-1)(2-1)
⇒r=R(2-1)2
Required Ratio = πr2πR2×2
=πR2(2-1)4×2πR2
=2(2-1)4:1
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