Points $A$ and $B$ lie $5$ units apart on the line $\ell .$ The point $P$ lies $6$ units away from $B$ such that $\mathrm{\angle }BPA={25}^{\circ }.$

1. Find the two possible values for the distance $AP.$

2. Find the shortest possible distance from $P$ to $\ell .$

For (a), you can use the cosine rule.

The situation is drawn below. Notice that there are two possible configurations, corresponding to the two different solutions you get from the cosine rule!

The diagram shows the first possible position of $P.$ Move around the point $P$ to find the other position where the angle at $P$ is ${25}^{\circ }.$

For (b), now you know the two values of $AP,$ you can use the sine rule to find some missing angles to help solve the problem.

Note: the shortest distance from $P$ to $\ell$ is perpendicular to $\ell .$