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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.
Solution
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.
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