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\includegraphics[max width=\textwidth, alt={}, center]{574b96b2-62f2-41b3-a178-8e68e16429ff-08_509_654_264_751}
The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius \(r\). The line \(B D\) is perpendicular to \(A C\). Angle \(C A B\) is \(\theta\) radians.
- Given that \(\theta = \frac { 1 } { 6 } \pi\), find the exact area of \(B C D\) in terms of \(r\).
- Given instead that the length of \(B D\) is \(\frac { \sqrt { 3 } } { 2 } r\), find the exact perimeter of \(B C D\) in terms of \(r\). [4]