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Chord AB subtends an angle of $120^\circ$ at the centre O of the circle with radius $\frac{21}{2}$ cm. Find the perimeter of shaded segment ACB. (Use $\sqrt{3} = 1.7$)
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Draw OD $\perp$ AB
$\therefore \angle AOD = 60^\circ$
$\sin 60^\circ = \frac{\sqrt{3}}{2} = \frac{AD}{\frac{21}{2}}$ (I) (1 Mark)
$\Rightarrow AD = \frac{21\sqrt{3}}{4}$ cm (II) (1/2 Mark)
AB = $2 \times AD = \frac{21\sqrt{3}}{2}$ cm
Also, length of minor arc AB $= \frac{120}{360} \times 2 \times \frac{22}{7} \times \frac{21}{2} = 22$ cm (III) (1 Mark)
$\therefore$ Perimeter of shaded region = $(22 + \frac{21\sqrt{3}}{2}) = 39.85$ cm (IV) (1/2 Mark)
$\therefore \angle AOD = 60^\circ$
$\sin 60^\circ = \frac{\sqrt{3}}{2} = \frac{AD}{\frac{21}{2}}$ (I) (1 Mark)
$\Rightarrow AD = \frac{21\sqrt{3}}{4}$ cm (II) (1/2 Mark)
AB = $2 \times AD = \frac{21\sqrt{3}}{2}$ cm
Also, length of minor arc AB $= \frac{120}{360} \times 2 \times \frac{22}{7} \times \frac{21}{2} = 22$ cm (III) (1 Mark)
$\therefore$ Perimeter of shaded region = $(22 + \frac{21\sqrt{3}}{2}) = 39.85$ cm (IV) (1/2 Mark)