Question

A circum-circle is drawn around a right triangle PQR. The height and the base of the triangle are 5 cm and 12 cm, respectively. What would be the area of the shaded region?

Answer 1

Here we are asked to find the area of the shaded region.

The area of the shaded region = Area of the circle – Area of the right triangle

The area of right triangle = $\frac{1}{2}\times height\times base$ = $\frac{1}{2}\times 12\times 5$ = 30 $c{m}^2$

Next we need to find the area of the circle, but to find the area we need to find the diameter or radius of the circle.

We know that, “if the angle subtended by a chord on the circumference is ${90}^{\circ }$, then the chord is the diameter of the circle”.

Therefore the hypotenuse of the triangle PQR will be diameter the circle. By applying the theorem we get,

$d^2=P{Q}^2+Q{R}^2$

$d^2=5^2+{12}^2$

d = 13 cm.

The area of the circle = $\pi \times (\frac{d}{2}{)}^2$ = $\pi \times (\frac{13}{2}{)}^2$ = 132.78 $c{m}^2$

Area of the shaded region = Area of the circle – Area of the triangle = 132.78 – 30 = 102.78 $c{m}^2$

The area of the shaded region is 102.78 $c{m}^2$ .