NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.3 – Part 2

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Conditions of congruence of triangles

Congruence Conditions

Conditions of congruence of triangles

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

Solution:

Circles Interact Each Other at a and B

Circles intersect at A and B, OO’ is perpendicular bisector of AB

  • 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)

  • (Side-Side-Side congruence condition)

Thus, (by corresponding parts of congruent triangle)

In ΔAOC and ΔBOC,

  • (Radius)

  • (Common line)

  • (Side-Angle-Side 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.