Question

The figure shows a rectangle ABCD with a semi circle and a circle inscribed inside it as shown. What is the ratio of the area of the circle to that of the semi circle?

• A. ${(\sqrt{2}-1)}^2$
• B. $2{(\sqrt{2}-1)}^2$
• C. ${(\sqrt{2}-1)}^4$
• D. None of these

Answer 1

The correct option is C ${(\sqrt{2}-1)}^4$

Let the centre be ${}^{\prime }{O}^{\prime }$ and the point at which the semicircle intersect $CD$ be $P$. Let the radius of the semi-circle be $R$ and the circle be ${}^{\prime }{r}^{\prime }$.

$OP=R$ and $OC=R\sqrt{2}$

$OC-OT=C{C}^1-T{C}^1\Rightarrow R\sqrt{2}-R-2r=r\sqrt{2}-r$

$\Rightarrow R\sqrt{2}-R=r\sqrt{2}+r\Rightarrow r=\frac{(\sqrt{2}-1)}{\sqrt{2}+1}R$

$\Rightarrow r={(\sqrt{2}-1)}^{2}R$

Ratio of areas $=r^2:R^2={(\sqrt{2}-1)}^4:1$