Tangent meets circle problems

Questions where tangents to a circle meet at an external point, requiring calculation of tangent lengths, angles, and shaded regions between tangents and arcs.

1 questions · Standard +0.3

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CAIE P1 2008 November Q6
8 marks Standard +0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_597_417_274_865} In the diagram, the circle has centre \(O\) and radius 5 cm . The points \(P\) and \(Q\) lie on the circle, and the arc length \(P Q\) is 9 cm . The tangents to the circle at \(P\) and \(Q\) meet at the point \(T\). Calculate
  1. angle \(P O Q\) in radians,
  2. the length of \(P T\),
  3. the area of the shaded region.