1.10: A sector and tangents to a circle

Figure 1 shows a sector $\displaystyle OPQ$ of a circle with centre $\displaystyle O$. The radius of the circle is $\displaystyle 18\,\mathrm{cm}$ and the angle $\displaystyle POQ$ is $\displaystyle \frac{2\pi}{3}$ radians.

(a) Find the length of the arc $\displaystyle PQ$, giving your answer as a multiple of $\displaystyle \pi$.

Figure 2 shows the sector $\displaystyle OPQ$ and the kite $\displaystyle OPTQ$. $\displaystyle PT$ is the tangent to the circle at $\displaystyle P$ and $\displaystyle QT$ is the tangent at $\displaystyle Q$, such that angle $\displaystyle PTQ=\alpha$ radians.

(b) (i) Find $\displaystyle \alpha$ in terms of $\displaystyle \pi$.

(ii) Calculate, to 3 significant figures, the area of the region, shown shaded in Figure 2, which is bounded by the arc $\displaystyle PQ$ and the tangents $\displaystyle PT$ and $\displaystyle QT$.