In a fig. a crescent is formed by two circles that touch at A; and C is the center of a larger circle. The width of the crescent at BD is 9 cm and at EF it is 5 cm. FC is perpendicular to AB, then find the radii of two circles.

Given in two circles,
BD = 9 cm
EF = 5 cm
Suppose that,
The radius of the smaller circle = r
The radius of larger circles = R

$\Rightarrow \mathrm{Diameter} \mathrm{of} \mathrm{Larger} \mathrm{Circle}-Diameter of Smaller Circle=BD$
$\Rightarrow 2\mathrm{R}-2\mathrm{r}=9$
$\Rightarrow 2(\mathrm{R}-\mathrm{r})=9$         

$\Rightarrow \mathrm{R}-\mathrm{r}=\frac{9}{2}$
$\Rightarrow \mathrm{R}-\mathrm{r}=4.5$ .............(i)
Now, we have joined the lines AE and DE.
We have, 
The angle of $\angle \mathrm{CAE}=\mathrm{\theta }$. 

Now, $\angle \mathrm{AED}={90}^{\circ }$   [Angle subtended by diameter of the smaller circle at its circumference]

$\angle \mathrm{AEC}=90°-\mathrm{\theta }$         [From triangle AEC]
So, we have
$\Rightarrow \angle \mathrm{AEC}+\angle \mathrm{DEC}={90}^{\circ }$
$\Rightarrow \angle \mathrm{DEC}={90}^{\circ }-\left({90}^{\circ }-\mathrm{\theta }\right)$
$\Rightarrow \angle \mathrm{DEC}=\mathrm{\theta }$
In the triangle $△s \mathrm{ACE}$ and$\text{ DCE}$, 

$$\begin{gathered} \angle \mathrm{DEC}=\angle \mathrm{EAC}=\mathrm{\theta } \\ \angle \mathrm{DCE}=\angle \mathrm{ACE}=90° \end{gathered}$$

by Angle - Angle similar criterion, we have;
$△\mathrm{ACE}\sim △\mathrm{ECD}$ 

From a side-ratio property of a similar triangle,
$\Rightarrow \frac{\mathrm{AC}}{\mathrm{EC}}=\frac{\mathrm{CE}}{\mathrm{CD}}$
$\Rightarrow \frac{\mathrm{AC}}{\mathrm{CF}-\mathrm{EF}}=\frac{\mathrm{CF}-\mathrm{EF}}{\mathrm{BC}-\mathrm{BD}}$

$\Rightarrow \frac{\mathrm{R}}{\mathrm{R}-5}=\frac{\mathrm{R}-5}{\mathrm{R}-9}$
$\Rightarrow \mathrm{R}(\mathrm{R}-9)=(\mathrm{R}-5{)}^2$

$\Rightarrow {\mathrm{R}}^2-9\mathrm{R}={\mathrm{R}}^2+25-10\mathrm{R}$
$\Rightarrow 0=-\mathrm{R}+25$
$\Rightarrow \mathrm{R}=25\mathrm{ cm}$ ...........(ii)

From (i) and (ii),
$\Rightarrow 25-\mathrm{r}=4.5$
$\Rightarrow \mathrm{r}=20.5 \mathrm{cm}$
Hence, the radii of two circles$\mathrm{ R}=25\mathrm{ cm}$ and$\mathrm{ }\mathrm{r}=20.5 \mathrm{cm}$.