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AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B. What is the relation between PQ and AB?

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Lengths of PQ and AB are equal

PQ is perpendicular bisector of AB

PQ is parallel to AB

PQ and AB are coincident lines

answer is B.

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Detailed Solution

Given, AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B.
Question Image

△ PAQ

and

△ PBQ

,
AP = BP (Since P is equidistant from A and B)
AQ = BQ (Since Q is equidistant from A and B)
PQ = PQ (Common)
So,

△ PAQ ⩭ △ PBQ

.[By SSS congruence rule]
Thus, ∠APQ = ∠BPQ.[Since, the two triangles are similar].
Now in

△ PAC

and

△ PBC

,
AP = BP (Since P is equidistant from A and B)
∠APC = ∠BPC (Since, AP = BP)
PC = PC (Common)
So,

△ PAC ⩭ △ PBC

.[By SAS congruence rule]
Therefore, AC = BC, ∠ACP = ∠BCP.[Since, the two triangles are similar]
Since, AB is a line segment, ∠ACP + ∠BCP = 180°.

\Rightarrow

2∠ACP = 180°

\Rightarrow

∠ACP = 90°
AC = BC, ∠ACP = ∠BCP = 90°.
Thus, PQ is a perpendicular bisector of AB.
Hence, option 2 is correct

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