Exercise 2 Page 328

Examining the given diagram, we can see that $\angle PSQ$ and $\angle QSR$ are adjacent congruent angles. Therefore, $\overrightarrow{SQ}$ forms a bisector of $\angle PSR.$ Moreover, the segments connecting $Q$ with the sides of $\angle PSR$ are both perpendicular to the sides. This means that their length is the distance from each side to $Q.$

According to the Angle Bisector Theorem, if a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle. In our case, since $Q$ lies on $\overrightarrow{SQ},$ we know that it is equidistant from the sides of $\angle PSR.$ Therefore, $\overline{QP}$ and $\overline{QR}$ have equal lengths. This allows us to equate the given expressions for their lengths to solve for $x.$ We found that $x=3,$ so let's substitute its value into the expression of $QP.$ Therefore, $QP=18.$