Chapter 4.7, Problem 23WE

To determine

To prove: $PQ=TS$ .

Explanation of Solution

Given information:

  $\overleftrightarrow{SR}$ is the perpendicular bisector of $\overline{QT}$ .

  $\overleftrightarrow{QR}$ is the perpendicular bisector of $\overline{SP}$ .

Given figure,

  

Calculation:

Since, $\overleftrightarrow{SR}$ is the perpendicular bisector of $\overline{QT}$ .

Therefore, point $S$ is equidistant endpoints $Q$ and $T$ of segment $\overline{QT}$ .

This gives $SQ=ST$

Also $\overleftrightarrow{QR}$ is the perpendicular bisector of $\overline{SP}$ , therefore, point $Q$ is equidistant from endpoints $S$ and $P$ of segment $\overline{SP}$ .

This gives $QP=QS$ .

Further, $SQ=ST$ and $QP=QS$

Therefore, $ST=QP$ or $PQ=TS$

Hence, proved.