NCERT Solutions Class 9 – Chapter 9 Circles – Exercise 9.1
Last Updated :
26 Apr, 2024
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.Â
Solution:
Given:Â
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. ∠AOB = ∠PXQ
Proof:
In â–³AOB & â–³PXQ
AO = PX Â Â Â Â Â Â Â Â Â Â (Radius of congruent circles are equal)
BO = QX Â Â Â Â Â Â Â Â Â Â Â (Radius of congruent circles are equal)
AB = PQ Â Â Â Â Â Â Â Â Â Â (Given)
â–³AOB â© â–³PXQ Â Â Â (SSS congruence rule)
Therefore, ∠AOB = ∠PXQ     (CPCT)
Question 2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Solution:Â
Given:
Two Congruent circles C1 and C2
AB is the chord of C1 and PQ is chord of C2
& ∠AOB = ∠PXQ Â
To prove :Â
In △AOB and  △PXQ ,
AO = PX Â Â Â Â Â Â Â Â Â Â Â Â (Radius of congruent circles are equal)
∠AOB = ∠PXQ        (Given)
BO = QX Â Â Â Â Â Â Â Â Â Â Â Â Â (Radius of congruent circles are equal)
â–³AOB â© â–³PXQ Â Â Â Â Â (SAS congruence rule)
Therefore, AB = PQ Â Â Â (CPCT)
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