Chapter 14.2, Problem 19WE

To determine

To prove that $PQ=P\text{'}Q\text{'}$

Explanation of Solution

Given information:

It is given that $\overline{P\text{'}Q\text{'}}$ is reflection of $PQ$ in line m .

Proof:

Since, $\overline{P\text{'}Q\text{'}}$ is reflection of $PQ$ in line m and it is visible in the figure that P and Q are not on the line m hence, m is perpendicular bisector of $PP\text{'}\text{and }QQ\text{'}$ , and if m is perpendicular bisector of both $PP\text{'}\text{and }QQ\text{'}$ . Since, $PP\text{'}\text{and}QQ\text{'}$ are not on same line then it obvious that $PP\text{'}\text{ and }QQ\text{'}$ are parallel.

The two statements that m is perpendicular bisector of $PP\text{'}$ and $QQ\text{'}$ , $PP\text{'}$ is parallel to $QQ\text{'}$ leads to conclude that $PQ=P\text{'}Q\text{'}$