Chord AB of a circle with centre O and radius $\displaystyle 21$ mm subtends an angle of $\displaystyle 120^{\mathrm{o}}$ at the centre. Find the perimeters of the shaded region. (Use $\displaystyle \sqrt{3}=1.73$ )

Draw OC ⟂ AB \(\displaystyle \therefore \angle \mathrm{AOC}=60^{0}\) \(\displaystyle \Rightarrow \sin 60^{0}=\frac{\sqrt{3}}{2}=\frac{A C}{21}\) \(\displaystyle \Rightarrow \mathrm{AC}=\frac{21 \sqrt{3}}{2}\) \(\displaystyle \Rightarrow \mathrm{AB}=2(\mathrm{AD})=21 \sqrt{3} \mathrm{~mm}\) Also, length of minor \(\displaystyle \operatorname{arc} \mathrm{AB}=\frac{120}{360} \times 2 \times \frac{22}{7} \times 21=44 \mathrm{~mm}\) ∴ Perimeter of shaded region \(\displaystyle =(44+21 \sqrt{3})=80.33 \mathrm{~mm}\)