Question

In the figure, AP and BQ are perpendiculars to the line segment AB and AP = BQ. Prove that O is the mid-point of the line segments AB and PQ.

Answer 1

Since AP and BQ are perpendiculars to the line segment AB, therefore Ap and BQ are parallel to each other.
In ΔAOP and ΔBOQ
∠PAQ = ∠QBO = 90°
∠APO = ∠BQO  ...(alternate angles)
AP = BQ
Therefore, ΔAOP ≅ ΔBOQ  AOP BOQ ...(ASA criteria)
Hence, AO = OB and PO = OQ
Thus, O is the mid-point of the line segments AB and PQ.