Arc Length and Sector Area

(CIE 0606/2021/m/22/Q6)
$A O B$ is a sector of a circle with centre $O$ and radius $16 \mathrm{~cm}$. Angle $A O B$ is $\dfrac{2 \pi}{7}$ radians. The point $C$ lies on $O B$ such that $O C$ is of length $7.5 \mathrm{~cm}$ and $A C$ is a straight line.

(a) Find the perimeter of the shaded region.

(b) Find the area of the shaded region.$[3]$


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$\begin{array}[t]{ll}\text{(a) }&\text { In } \triangle \text { OAC, by cosine rule, } \\&A C=\sqrt{16^{2}+7.5^{2}-2(16)(7.5) \cos \left(\dfrac{2 \pi}{7}\right)}=12.75 \\&\text { are length } A B=16\left(\dfrac{2 \pi}{7}\right)=14.36 \\&\text { Hence the perimeter of shaded region } \\&=A C+(\text { arc length } A B)+B C \\&=12.75+14.36+(16-705)=35.6 \\\text{(b) }&\text { Area of } \triangle O A C=\dfrac{1}{2} \times 16 \times 7.5 \times \sin \left(\frac{2 \pi}{7}\right)=46.909 \\&\therefore \text { Area of shaded region } =\dfrac{1}{2} \times 16^{2} \times\left(\dfrac{2 \pi}{7}\right)-46.909=68.0\end{array}$
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