NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.3 – Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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
  • 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.