Question

ab is a line segment p and q are points on the opposite sides of ab such that each of them is equidistant from point a and b show that line pq is perpendicular bisector of ab

Answer 1

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Answer 2

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Given P is equidistant from points A and B

PA=PB .....(1)

and Q is equidistant from points A and B

QA=QB .....(2)

In △PAQ and △PBQ

AP=BP from (1)

AQ=BQ from (2)

PQ=PQ (common)

So, △PAQ≅△PBQ (SSS congruence)

Hence ∠APQ=∠BPQ by CPCT

In △PAC and △PBC

AP=BP from (1)

∠APC=∠BPC from (3)

PC=PC (common)

△PAC≅△PBC (SAS congruence)

∴AC=BC by CPCT

and ∠ACP=∠BCP by CPCT ....(4)

Since, AB is a line segment,

∠ACP+∠BCP=180

(linear pair)

∠ACP+∠ACP=180

from (4)

2∠ACP=180

∠ACP=

2

180

=90

Thus, AC=BC and ∠ACP=∠BCP=90

∴,PQ is perpendicular bisector of AB.

Hence proved