Class 9 NCERT Solutions – Chapter 10 Circles – Exercise 10.2

Question 1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

Solution:

Given:

Two Congruent Circles C1 and C2

AB is the chord of C1

and PQ is the chord of C2

AB = PQ

To Prove: Angle subtended by the Chords AB and PQ are equal i.e. âˆ AOB = âˆ PXQ

Proof:

In â–³AOB & â–³PXQ

AO = PX                    (Radius of congruent circles are equal)

BO = QX                      (Radius of congruent circles are equal)

AB = PQ                    (Given)

â–³AOB â©­ â–³PXQ       (SSS congruence rule)

Therefore, âˆ AOB = âˆ PXQ         (CPCT)

Question 2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Solution:

Given:

Two Congruent circles C1 and C2

AB is the chord of C1 and PQ is chord of C2

& âˆ AOB = âˆ PXQ

To prove :

In â–³AOB and  â–³PXQ ,

AO = PX                         (Radius of congruent circles are equal)

âˆ AOB = âˆ PXQ              (Given)

BO = QX                          (Radius of congruent circles are equal)

â–³AOB â©­ â–³PXQ          (SAS congruence rule)

Therefore, AB = PQ       (CPCT)

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