Problem #238
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238
(a) In Fig. 1, \(OAB\) is a sector of a circle with centre \(O\) and radius \(r\). \(AX\) is the tangent at \(A\) to the arc \(AB\) and angle \(BAX = \alpha\).
(i) Show that angle \(AOB = 2\alpha\).
(ii) Find the area of the shaded segment in terms of \(r\) and \(\alpha\).
(b) In Fig. 2, \(ABC\) is an equilateral triangle of side 4 cm. The lines \(AX, BX\) and \(CX\) are tangents to the equal circular arcs \(AB, BC\) and \(CA\). Use the results in part (a) to find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt{3}\).
