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  4. 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 <img class=img-fluid question-image alt=Question src=https://cdn.tardigrade.in/q/nta/m-uub6b3yf484imag4.jpg />

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Q. $ABC$ 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
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NTA AbhyasNTA Abhyas 2022

Solution:

From the given figure, $\triangle PRB$ 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 $=4\pi -\left(\right.\pi -2$ )
$=3\pi +2$ sq. units