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In the diagram, \(A B\) is an arc of a circle with centre \(O\) and radius \(r\). The line \(X B\) is a tangent to the circle at \(B\) and \(A\) is the mid-point of \(O X\).
- Show that angle \(A O B = \frac { 1 } { 3 } \pi\) radians.
Express each of the following in terms of \(r , \pi\) and \(\sqrt { } 3\) :
- the perimeter of the shaded region,
- the area of the shaded region.