Question

jamie is proving that any point on the perpendicular bisector of a line segment is equidistant of the line segment. he is using the figure shown. he knows that line ms is the perpendicular bisector of $overline{pq}$. his proof is shown.

statement	reason
2. draw $overline{sp}$ and $overline{qs}$	2. there exists a line through any 2 points
3. $overline{mp}=overline{mq}$	3. definition of bisector
4. $overline{ms}=overline{ms}$	4. reflexive property

Explanation:

Step1: Identify given information

Given that $MS$ is the perpendicular bisector of $\overline{PQ}$.

Step2: Recall geometric properties

By the definition of a bisector, $\overline{MP}=\overline{MQ}$. By the reflexive - property, $\overline{MS}=\overline{MS}$.

Step3: Consider right - triangle congruence

Since $MS$ is perpendicular to $PQ$, $\angle PMS=\angle QMS = 90^{\circ}$. We can use the Side - Angle - Side (SAS) congruence criterion for $\triangle PMS$ and $\triangle QMS$. In $\triangle PMS$ and $\triangle QMS$, we have $\overline{MP}=\overline{MQ}$, $\angle PMS=\angle QMS$, and $\overline{MS}=\overline{MS}$. So, $\triangle PMS\cong\triangle QMS$.

Step4: Use congruent - triangle properties

If $\triangle PMS\cong\triangle QMS$, then corresponding sides are equal. So, $\overline{SP}=\overline{SQ}$, which shows that any point $S$ on the perpendicular bisector of a line segment $PQ$ is equidistant from the endpoints $P$ and $Q$.

Answer:

The proof shows that any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment.