Question

In figure, AP and BQ are perpendiculars to the line segment AB and AP = BQ. Prove that O is the midpoint of line segment AB as well as PQ.

Answer 1

Given: AP = BQ and OAP = OBQ = 90°

To Prove: AO = OB and PO = OQ

Proof:

In ΔAOP and ΔBOQ:

∠OAP = ∠OBQ (Given)

∠AOP = ∠BOQ (Vertically opposite angles)

AP = BQ (Given)

ΔAOP ΔBOQ (AAS congruency)

∴ AO = OB and PO = QO (Corresponding parts of congruent triangles)

Thus, O is the midpoint of AB and PQ