Question

In figure $AB=36cm$ and $M$ is mid-point of $AB$. Semicircles are drawn on $AB$, $AM$ and $MB$ as diameter. A circle with centre $C$ touches all the three circles. Find the area of the shaded region.

• A. $167.1c{m}^2$
• B. $100.1c{m}^2$
• C. $341.1c{m}^2$
• D. $141.1c{m}^2$

Answer 1

The correct option is D $141.1c{m}^2$

In $△COM$

$(OC{)}^2=(CM{)}^2+(OM{)}^2$

$(x+9{)}^2=(18-x{)}^2+9^2$

$x^2+8x+18r=(18{)}^2+x^2-36x+81$

$54r=18\times 18$

$r=\frac{18\times 18}{54}$

$x=6$

Area of shaded region $\Rightarrow$

$\frac{\pi (AM{)}^2}{2}-\frac{\pi (OA{)}^2}{2}-\frac{\pi (QM{)}^2}{2}-\pi (CP{)}^2$

$\frac{\pi (18{)}^2}{2}-\frac{\pi (9{)}^2}{2}-\frac{\pi (9{)}^2}{2}-\pi (6{)}^2$

$162\pi -81\pi -36\pi$

$=45\pi$

$=141.3c{m}^2$