Question

$ABC$ is a right triangle, right-angled at the vertex $A.$ A circle is drawn to touch the sides $AB$ and $AC$ at points $P$ and $Q$ respectively such that the other end points of the diameters passing through $P$ and $Q$ lie on the side $BC.$ If $AB=6$ units, then the area (in sq. units) of the circular sector which lies inside the triangle is

• A. $3\pi +2$
• B. $\pi +3$
• C. $\pi -2$
• D. $\pi +2$

Answer 1

Answer: (A)

From the given figure, $△PRB$ is an isoceles triangle.
$PB=PR=2r$
$AB=3r=6\Rightarrow r=2$
Shaded area $=\frac{1}{4}\pi {\left(2\right)}^2-\frac{1}{2}2.2=\pi -2$
Required area $=4\pi -(\pi -2$ )
$=3\pi +2$ sq. units