9. The diagram shows two concentric circles. Chord $A B$ of the larger circle is tangential to the smaller circle.
The length of $A B$ is $32 \mathrm{~cm}$ and the area of the shaded region is $k \pi \mathrm{cm}^{2}$.
What is the value of $k$ ?
9. 256 Let the radii of the larger and smaller circles be $R$ and $r$ respectively. Draw radius $O A$ of the larger circle and drop the perpendicular from $O$ to $A B$. By the tangent-radius property this perpendicular will be a radius of the smaller circle.
Now the area of the shaded region $=$ area of larger circle area of smaller circle.
The area of the shaded region $=\pi R^{2}-\pi r^{2}=\pi\left(R^{2}-r^{2}\right)$. But $R^{2}-r^{2}=16^{2}=256$ (by Pythagoras' theorem), hence the area of the shaded region $=256 \pi$ and therefore $k=256$.