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Question

Prove that the line joining the centres of two intersecting circles is the perpendicular bisector of the line joining the points of intersection.

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Solution

Let A and B be centres of two circles intersecting at points P and Q.
In APB and AQB,
AP=AQThey are radii of a circle.
BP=BQThey are radii of a circle.
AB=ABCommon side
APBAQB By S.S.S. Criterion
PAB=QAB By C.P.C.T.C.
Now, In APR and AQR
AP=AQ They are radii of a circle.
PAR=QARPAB=QAB
AR=AR Common side
APRAQR ....... By S.A.S. Criterion
PR=RQ ..... By CPCT
And ARP=ARQ ....... By C.P.C.T.C.
Also, ARP+ARQ=180o PQ is a straight line.
ARP+ARP=180o
2×ARP=180o
ARP=180o2=90o
Thus, ABrPQ

988537_1084756_ans_364d7f117cb4433a9357393031528df7.png

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