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\includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-3_579_659_269_744}
In the diagram, \(A B\) is an arc of a circle, centre \(O\) and radius \(r \mathrm {~cm}\), and angle \(A O B = \theta\) radians. The point \(X\) lies on \(O B\) and \(A X\) is perpendicular to \(O B\).
- Show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the shaded region \(A X B\) is given by
$$A = \frac { 1 } { 2 } r ^ { 2 } ( \theta - \sin \theta \cos \theta ) .$$
- In the case where \(r = 12\) and \(\theta = \frac { 1 } { 6 } \pi\), find the perimeter of the shaded region \(A X B\), leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).