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Question

AB is a line segment C and D are points on opposite sides of AB such that each of them is equidistant from the point A and B as in the figure. The line CD is perpendicular bisector of AB.
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A
True
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B
False
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Solution

The correct option is A True
C is equidistant from points A and B
CA=CB and D is equidistant from points A and B
DA=DB
We have to prove that CDAB
AD=BD and CPA=CPB=90
In CAD and CBD
AC=BC ....... (1)
AD=BD ........ (2)
CD=CD ......... (common)
CADCBD ..... (SSS congruence rule)
Hence ACD=BCD ........ (C.P.C.T) (3)
In CAPCBP ........ (SAS congruence rule)
AP=BP(C.P.C.T)
APC=BPC ........ (C.P.C.T)
Since AB is a line segment
APC+BPC=180
2APC=180APC=90
APC=BPC=90
Hence, AC=BC and APC=BPC=90
CD is bisector of AB

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