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\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-2_330_634_753_758}
In the diagram, \(O P Q\) is a sector of a circle, centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(Q O P = \theta\) radians. The tangent to the circle at \(Q\) meets \(O P\) extended at \(R\).
- Show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the shaded region is given by \(A = \frac { 1 } { 2 } r ^ { 2 } ( \tan \theta - \theta )\).
- In the case where \(\theta = 0.8\) and \(r = 15\), evaluate the length of the perimeter of the shaded region.