Question

Two semi-circles are drawn on the diameter of a semi-circle with radius $18cm$. A circle with centre C is drawn such that it touches the two semi-circles. Find area of shaded region.

Answer 1

Draw $RM\perp AB$

$AM=MB=\frac{AB}{2}=18cm$

$AP=PM=MQ=QB=\frac{18}{2}=9cm$

$MR=AM=18㎝$

$CM=RM-CR=18-r$

$PC=PE+EC=9+r$

In triangle $CMP$

${PC}^2={CM}^2+{PM}^2\Rightarrow {(9+r)}^2={(18-r)}^2+9^2$

$\Rightarrow 81+18r+r^2=324-36r+r^2+81\Rightarrow 54r=324\Rightarrow r=6cm$

$\therefore$Area of shaded region=area of semi circle AB$-2$ area of semi circle-area of circle with C as centre

$=\frac{1}{2}\pi \times {18}^2-2\times \frac{1}{2}\times \pi \times 9^2-\pi \times 6^2$

$=162\pi -81\pi -36\pi =45\pi c{m}^2$