Problem Statement
A circle of radius $r = 6\text{ cm}$ has a chord $AB$ subtending an angle of $\theta = \frac{2\pi}{3}$ radians at the center $O$.
**(a)** [2 marks] Find the exact length of the minor arc $AB$.
**(b)** [4 marks] Find the exact area of the shaded minor segment bounded by the chord $AB$ and the minor arc $AB$.
Verified Solution & Marking Scheme
Part (a) [M1]
Arc length formula: $l = r\theta$
Part (a) [A1]
$l = 6\left(\frac{2\pi}{3}\right) = 4\pi\text{ cm}$
Part (b) [M1]
Sector area: $A_{\text{sector}} = \frac{1}{2}r^2\theta = \frac{1}{2}(6^2)\left(\frac{2\pi}{3}\right) = 18\left(\frac{2\pi}{3}\right) = 12\pi$
Part (b) [M1]
Triangle area: $A_{\triangle OAB} = \frac{1}{2}r^2\sin\theta = \frac{1}{2}(36)\sin\left(\frac{2\pi}{3}\right) = 18\left(\frac{\sqrt{3}}{2}\right) = 9\sqrt{3}$
Part (b) [M1]
Segment area $= A_{\text{sector}} - A_{\triangle OAB}$
Part (b) [A1]
$\text{Area} = 12\pi - 9\sqrt{3} = 3(4\pi - 3\sqrt{3})\text{ cm}^2$