11. A ball is propelled from corner $A$ of a square snooker table of side 2 metres. After bouncing off three cushions as shown, the ball goes into a pocket at $B$. The total distance travelled by the ball is $\sqrt{k}$ metres. What is the value of $k$ ?
(Note that when the ball bounces off a cushion, the angle its path makes with the cushion as it approaches the point of impact is equal to the angle its path makes with the cushion as it moves away from the point of impact as shown in the diagram below.)
11. 52 A solution can be obtained by reflecting the square repeatedly in the cushion the ball strikes. The path of the ball is then represented by the line $A B^{\prime}$ in the diagram.
The length of the path can be calculated using Pythagoras Theorem. We have $\left(A B^{\prime}\right)^{2}=(3 \times 2)^{2}+(2 \times 2)^{2}$. Therefore $\left(A B^{\prime}\right)^{2}=36+16=52$ and so $A B^{\prime}=\sqrt{52}$ metres and hence the value of $k$ is 52 .