Question

A circle is drawn in a sector of a larger circle of radius $r$ as shown in figure. The smaller circle is tangent to the two bounding radii and the arc of the sector. The radius of the small circle is

• A. $\frac{r}{2}$
• B. $\frac{r}{3}$
• C. $\frac{2\sqrt{3r}}{5}$
• D. $\frac{r}{\sqrt{2}}$

Answer 1

Answer: (B)

$\frac{r}{3}$

From the given figure.
Radius of the larger circle$=AC=r$
Radius of the smaller circle$=OB=BC=R$
$AB=AC-BC=r-R$
$\angle OAB=\frac{\angle OAD}{2}={30}^{}$
In right angled triangle ABO
$\sin \left(\angle OAB\right)=\frac{OB}{AB}$
$\sin {30}^{\circ }=\frac{R}{r-R}\Rightarrow R=\frac{r}{3}$