Question

In the given figure, $△PRT$ is an equilateral triangle. Line segments TQ and PS are perpendicular bisectors to side PR and side TR respectively. Which of these statement(s) is/are correct?

• A. $△PQT\cong △RQT$
• B. $△TSP\cong △RSP$
• C. $△RSP\cong △PQT$
• D. None of the above.

Answer 1

Answer: (A), (B), (C)

The correct options are
A $△PQT\cong △RQT$
B $△TSP\cong △RSP$
C $△RSP\cong △PQT$

Given, $△PRT$ is an equilateral.

So, PR = RT = TP

TQ and PS are perpendicular bisectors to the sides PR and TR respectively.

So, PQ = RQ = TS = SR

Therefore, in triangles,
$△PQT\text{and}△RQT,$
side PQ = side RQ
side QT is common
side TP = side TR

Hence, by SSS criterion for congruency of triangles,
$△PQT\cong △RQT.$

Again, in triangles,
$△TSP\text{and}△RSP,$
side TS = side RS
side SP is common
side PT = side PR

Hence, by SSS criterion for congruency of triangles,
$△TSP\cong △RSP.$

Also, in triangles,
$△RSP\text{and}△PQT,$
side RS = side PQ
$\angle RSP=\angle PQT={90}^{\circ }$
side PR = side TP (hypotenuse of the respective triangles)

Hence, by RHS criterion for congruency of triangles,
$△PQT\cong △RQT.$