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Chord AB of a circle with centre O and radius $21$ mm subtends an angle of $120^\circ$ at the centre. Find the perimeters of the shaded region. (Use $\sqrt{3} = 1.73$)
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Draw $OC \perp AB$
$\therefore \angle AOC = 60^\circ$
$\sin 60^\circ = \frac{AC}{21} = \frac{\sqrt{3}}{2}$ (1 Mark)
$\Rightarrow AC = \frac{21\sqrt{3}}{2}$
$\Rightarrow AB = 2 (AD) = 21\sqrt{3}$ mm (1/2 Mark)
Also, length of minor arc AB = $\frac{120}{360} \times 2 \times \frac{22}{7} \times 21 = 44$ mm (1 Mark)
$\therefore$ Perimeter of shaded region = $(44 + 21\sqrt{3}) = 80.33$ mm (1/2 Mark)
$\therefore \angle AOC = 60^\circ$
$\sin 60^\circ = \frac{AC}{21} = \frac{\sqrt{3}}{2}$ (1 Mark)
$\Rightarrow AC = \frac{21\sqrt{3}}{2}$
$\Rightarrow AB = 2 (AD) = 21\sqrt{3}$ mm (1/2 Mark)
Also, length of minor arc AB = $\frac{120}{360} \times 2 \times \frac{22}{7} \times 21 = 44$ mm (1 Mark)
$\therefore$ Perimeter of shaded region = $(44 + 21\sqrt{3}) = 80.33$ mm (1/2 Mark)