Question

If ABC is a quarter circle and a circle is inscribed in it and if AB = 1 cm, find the radius of the smaller circle.

• A. $\sqrt{2}-1$
• B. $\sqrt{2}+1)/2$
• C. $\sqrt{2}-1/2$
• D. $1-2\sqrt{2}$

Answer 1

The correct option is A $\sqrt{2}-1$

Area of the quarter circle $\frac{\pi r^2}{4}\Rightarrow 0.25\pi$. Going by options, we have to see that the area of the inserted circle is less than the area of the quarter circle.
option (b) $\frac{(\sqrt{2}+1)}{2}\Rightarrow Area=(1.5+\sqrt{2})\pi$
$2.9\pi >0.25\pi .$ Hence discarded.
Option (c) $\sqrt{2}-\frac{1}{2}\Rightarrow Area=\pi (2+\frac{1}{4}-\sqrt{2})\Rightarrow 0.75\pi$
$>0.25\pi .$
Hence discarded.
Option (d) $1-\sqrt[2]{22}\Rightarrow Area=\pi (1+8-\sqrt[4]{42})\Rightarrow 0.85\pi$
$>0.25\pi .$
Hence discarded.
Option (a) $\sqrt{2}-1\Rightarrow Area\pi (2+1\sqrt[2]{22})\Rightarrow 0.20\pi$
$<0.25\pi$
Hence this option is correct.