In given figure AB is a line-segment. P and Q are points on either side of AB such that each of them is equidistant from the points A and B. Show that the line PQ is the perpendicular bisector of AB
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Solution
Given AB is a line segment .P and Q are two-point on either side of AB such That AP=BPandAQ=BQ
In ΔPAQand ΔPBQ
AP=BP (Given )
AQ=BQ (Given )
PQ=PQ (Common side)
∴ΔPAQ≅ΔPBQ
∴∠APQ=∠BPQ
In ΔPAC and PBC
AP=BP (Given )
∠APC=∠BPC We proved above ∠APQ=∠BPQ
PC=PC (Common Side)
∴ΔPAC≅PBC
∴AC=BC....................................(1)
∴∠ACP=∠BCP
∠ACP+∠BCP=1800
∴2∠ACP=1800
⇒∠ACP=900.....................(2)
From (1) and (2) the line PQ is the perpendicular bisector of AB. [henceproved]