Chapter 6.5, Problem 2CYP

To determine

To prove: $PQRS$is a square.

Explanation of Solution

Given Information:

  $\overline{SQ}$ is the perpendicular bisector of $\overline{PR}$ .

  $\overline{PR}$ is the perpendicular bisector of $\overline{SQ}$ .

  $\Delta RMS$ is isosceles.

Prove:

Consider the given figure

  

It is given that $\overline{SQ}$ is the perpendicular bisector of $\overline{PR}$ .

  $\overline{PR}$ is the perpendicular bisector of $\overline{SQ}$ .

  $\Delta RMS$ is isosceles.

Thus, $\angle MSR=\angle MRS$

Since $PQ\Vert SR$

Thus, $\angle MSR=\angle MQP$

So, $\angle MSR=\angle MRS=\angle MQP$

Therefore, $\angle MSR=\angle MRS=\angle MQP=\angle MPQ=\angle MSP=\angle MPS=\angle MRQ=\angle MQR$

Thus, $\angle P=\angle S=\angle R=\angle Q$ .

Similarly, $\overline{SQ}$ and $\overline{PR}$ are the diagonal of the quadrilateral and both are perpendicular to each other.

Thus, the side $PS=SR=RQ=QP$

Thus, $PQRS$is a square.

Hence, proved.