Question

The diagram shows a right-angled triangle and a shaded semicircular region whose diameter is $BC$. The area of the shaded region is

• A. $4\pi {\text{cm}}^2$
• B. $2\pi {\text{cm}}^2$
• C. $8\pi {\text{cm}}^2$
• D. $16\pi {\text{cm}}^2$

Answer 1

The correct option is B $2\pi {\text{cm}}^2$


$\Delta ABC$ is right-angled at $B$.
$\therefore A{C}^2=A{B}^2+B{C}^2$
(By Pythagoras Theorem)
$\Rightarrow 5^2=3^2+B{C}^2$
$\Rightarrow B{C}^2=25–9=16$
$\Rightarrow BC=4\text{cm}$
$\therefore$ Area of semicircle with $BC$ as diameter
$=\frac{1}{2}\pi {\left(\frac{BC}{2}\right)}^2$
$=\frac{\pi }{8}\times (4{)}^2=2\pi {\text{cm}}^2$
Hence, the correct answer is option b.