PQRS is a quadrilateral in which PQ = PS and RQ = RS. Which of the following…

Concept: The set of points that are equidistant from two fixed points A and B is the perpendicular bisector of AB. So if two distinct points are each equidistant from A and B, the line joining those two points is the perpendicular bisector of AB: it meets AB at a right angle and cuts it into two equal parts.

Application to PQRS:

1. PQ = PS tells us that P is equidistant from Q and S, so P lies on the perpendicular bisector of the segment QS.

2. RQ = RS tells us that R is equidistant from Q and S, so R lies on that same perpendicular bisector of QS.

3. P and R are two distinct points, and two distinct points determine exactly one line, so the whole line PR is the perpendicular bisector of QS.

4. Hence the diagonal PR meets the diagonal QS at 90 degrees, and PR cuts QS into two equal halves.

Cross-check with a concrete example: take QS = 24 cm with PQ = PS = 13 cm and RQ = RS = 15 cm, and let M be the point where PR crosses QS, so that QM = MS = 12 cm.

• Diagonal lengths: triangle PMQ is right angled at M and uses the triple 5, 12, 13, so PM = 5 cm; triangle RMQ uses the triple 9, 12, 15, so MR = 9 cm. That gives PR = 14 cm against QS = 24 cm, so the two diagonals need not have the same length.

• Cutting each other in half: M divides QS into 12 cm and 12 cm, but it divides PR into 5 cm and 9 cm, so only one of the two diagonals is bisected.

• Angles at the vertices: reflecting the figure in the line PR maps Q onto S, so PR does bisect the angles at P and at R; at Q, however, the two parts of the angle have tangents 5/12 and 9/12, so QS is not an angle bisector there.

Result: For every quadrilateral with PQ = PS and RQ = RS the two diagonals cross at right angles. The other three properties need extra conditions, such as opposite sides being parallel or all four sides being equal, which the given data does not supply.