If ABC is a quarter circle and a circle is inscribed in it and if AB = 1 cm, find radius of smaller circle.
$\sqrt{2} - 1$
$(\sqrt{2 + 1})2$
$\sqrt{2} - 1/2$
$1 - 2\sqrt{2}$
None of these
Solution
Area of the quarter circle $= \dfrac{\pi r^2}{4} \Rightarrow 0.25 \pi$.
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) $\dfrac{(\sqrt{2} + 1)}{2} \Rightarrow$ Area = $(1.5 + \sqrt{2})\pi$
$2.9\pi > 0.25\pi$. Hence discarded.
Option (c) $\sqrt{2} - \dfrac{1}{2} \Rightarrow$ Area $= \pi \left( 2 + \dfrac{1}{4} - \sqrt{2} \right) \Rightarrow 0.75\pi > 0.25\pi$.
Hence discarded.
Option (d) $1 - 2\sqrt{2} \Rightarrow$ Area $= \pi(1 + 8 - 4\sqrt{2}) \Rightarrow 0.85\pi > 0.25\pi$.
Hence discarded.
Option (a) $\sqrt{2} - 1 = $ Area $\pi (2 + 1 - 2\sqrt{2}) \Rightarrow 0.20 \pi < 0.25\pi$
Hence this option is correct.