# Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection

**Solution:**

Draw two intersecting circles with centers O and O’, respectively. Join these two centers. Let the points of intersection be A and B.

We need to prove that ∠OAO'= ∠OBO'

Consider ΔOAO’ and ΔOBO’

OA = OB (Radii of a circle with center O)

O’A = O’B (Radii of a circle with center O’)

OO’= OO’ (Common)

Therefore, by SSS criteria, ΔOAO’ and ΔOBO’ are congruent to each other.

By CPCT, ∠OAO'= ∠OBO'

Hence it is proved that the line of centers of two intersecting circles subtends equal angles at the two points of intersection.

**☛ Check: **NCERT Solutions Class 9 Maths Chapter 10

**Video Solution:**

## Prove that the line of centers of two intersecting circles subtends equal angles at the two points of intersection

Maths NCERT Solutions Class 9 - Chapter 10 Exercise 10.6 Question 1:

**Summary:**

We have proved that the line of centers of two intersecting circles subtends equal angles at the two points of intersection.

**☛ Related Questions:**

- Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its center. If the distance between AB and CD is 6 cm, find the radius of the circle.
- The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the center, what is the distance of the other chord from the center ?
- Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the center.
- Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.

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