NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.3 – Part 2
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Q3 If two circles intersect at two points; prove that their centres lie on the perpendicular bisector of the common chord.
Solution:

A and B are the two points at the intersection of two circles.

To prove, AB is bisector and OO’ is perpendicular to OO’.
In and ,

(Radius)

(Common line)

(Radius)

(SideSideSide congruence condition)
Thus, (by corresponding parts of congruent triangle)
In ΔAOC and ΔBOC,

(Radius)


(Common line)

(SideAngleSide congruence condition)
Thus, (by corresponding parts of congruent triangles)
Also,


()


Hence, OO' is perpendicular to AB. Since , therefore AC = CB, i.e. C is the midpoint of AB. Therefore, OO' is perpendicular bisector of AB.