Question

In the given figure, a semicircle is drawn with $AB$ as diameter of length $14\text{cm}$ and $DO\perp AB$. $CD$ and $CB$ are tangents at $D$ and $B$ respectively and $OD=DC$. The area of shaded region is

• A. $11.6{\text{cm}}^2$
• B. $8.4{\text{cm}}^2$
• C. $13.4{\text{cm}}^2$
• D. $10.5{\text{cm}}^2$

Answer 1

The correct option is D $10.5{\text{cm}}^2$

Diameter of semicircle $=14\text{cm}$
$\Rightarrow$ Radius of semicircle, $OB=\frac{14}{2}=7\text{cm}$
$\therefore$ Area of quadrant $BOD=\frac{1}{4}\times \pi r^2$
$=\frac{1}{4}\times \frac{22}{7}\times 7\times 7$
$=38.5{\text{cm}}^2$

Since, $CD$ and $CB$ are tangents at $D$ and $B$ and $OD=DC$.
$\therefore CD=CB$
Also, $OD=OB$ (Radii)
And, $OD=CD$ (Given)
$\therefore OD=DC=CB=OB=7\text{cm}$
Thus, $BODC$ is a square.

Area of square $BODC={\text{(Side)}}^2=7^2=49{\text{cm}}^2$

$\therefore$ Area of shaded region = Area of square $BODC–$ Area of quadrant $BOD$
$=49–38.5$
$=10.5{\text{cm}}^2$

Hence, the correct answer is option d.