I want to know the best answer of the question from Circles chapter of class 9^{th} ncert math. The question from exercise 10.3 of math. Give me the easy way for solving this question of 3 If two circles intersect at two points, prove that their centers lie on the perpendicular bisector of the common chord.

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# If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord. Q.3

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It is given that two circles intersect each other at P and Q.

To prove:OO’ is perpendicular bisector of PQ.

Proof:Triangle ΔPOO’ and ΔQOO’ are similar by SSS congruency since

OP = OQ and O’P = OQ (Since they are also the radii)

OO’ = OO’ (It is the common side)

So, It can be said that ΔPOO’ ΔQOO’

∴ POO’ = QOO’ — (i)

Even triangles ΔPOR and ΔQOR are similar by SAS congruency as

OP = OQ (Radii)

POR = QOR (As POO’ = QOO’)

OR = OR (Common arm)

So, ΔPOR ΔQOR

∴ PRO = QRO

Also, we know that

PRO+QRO = 180°

Hence, PRO = QRO = 180°/2 = 90°

So, OO’ is the perpendicular bisector of PQ.