Chord AB subtends an angle of $\displaystyle 120$° at the centre O of the circle with radius $\displaystyle \frac{21}{2} \mathrm{~cm}$. Find the perimeter of shaded segment ACB. (Use $\displaystyle \sqrt{3}=1.7$ )

Draw \(\displaystyle \mathrm{OD} \perp \mathrm{AB}\) \(\displaystyle \therefore \angle \mathrm{AOD}=60^{0}\) \(\displaystyle \Rightarrow \sin 60^{0}=\frac{\sqrt{3}}{2}=\frac{\mathrm{AD}}{\dfrac{21}{2}}\) \(\displaystyle \Rightarrow \mathrm{AD}=\frac{21 \sqrt{3}}{4} \mathrm{~cm}\) \(\displaystyle \Rightarrow \mathrm{AB}=2 \times \mathrm{AD}=\frac{21 \sqrt{3}}{2} \mathrm{~cm}\) Also, length of minor arc \(\displaystyle \mathrm{AB}=\frac{120}{360} \times 2 \times \frac{22}{7} \times \frac{21}{2}=22 \mathrm{~cm}\) ∴ Perimeter of shaded region \(\displaystyle =\left(22+\frac{21 \sqrt{3}}{2}\right)=39.85 \mathrm{~cm}\)