Chapter 13.2, Problem 7MC

Andrea, a civil engineer, was asked to make a plan for the shortest possible path connecting a landmark P and two roads on $\stackrel{\to }{AB}$ and $\stackrel{\to }{AC}$. She needed to find point X on $\stackrel{\to }{AB}$ and point Y on $\stackrel{\to }{AC}$ so that the perimeter of $\Delta PXY$ is as short as possible. She wanted to explain her approach to her teenage daughter. Andrea first drew arbitrary points $X^{\prime }$ on $\stackrel{\to }{AB}$ and $Y^{\prime }$ on $\stackrel{\to }{AC}$ and then $\Delta X^{\prime }P{Y}^{\prime }$ for one possible path along the sides of the triangle, but not necessarily the shortest. Then she found $P^{\prime }$, the reflection of P in $\stackrel{\to }{AB}$ and $P^{″}$ the reflection of P in $\stackrel{\to }{AC}$, and claimed that the perimeter of the triangle equals the length of the path $P^{\prime }{X}^{\prime }{Y}^{\prime }{P}^{″}$. She then connected $P^{\prime }$ with $P^{″}$ by a straight line and found the intersection points X and Y with $\stackrel{\to }{AB}$ and $\stackrel{\to }{AC}$, respectively, claiming that the path along the sides of $\Delta XPY$ is the required shortest path.

Answer the following.

a. Why the perimeter of $\Delta X^{\prime }P{Y}^{\prime }$ is equal to the length of the path $P^{\prime }{X}^{\prime }{Y}^{\prime }{P}^{″}$?

b. Why the length of the path $P^{\prime }XPY{P}^{″}$ is equal to $P^{\prime }{P}^{″}$?

c. Why the perimeter of $\Delta XPY$ shorter than the perimeter of $\Delta X^{\prime }P{Y}^{\prime }$? Did Andrea find the shortest possible path? Explain.