Class 9 Circles Exercise 9.1 Solutions
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Overview
Review Class 9 Circles Exercise 9.1 with a worked proof about equal chords in congruent circles and the angles they subtend at the centres. Follow the congruence reasoning step by step.
Chords and Angles in Congruent Circles
Q1: Equal Chords & Angles
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.
Given: Two congruent circles with centres $O$ and $P$.
Chords $AB = CD$.
Chords $AB = CD$.
To Prove: $\angle AOB = \angle CPD$.
Proof: In $\triangle AOB$ and $\triangle CPD$:
1. $OA = PC$ (Radii of congruent circles)
2. $OB = PD$ (Radii of congruent circles)
3. $AB = CD$ (Given)
$\therefore \triangle AOB \cong \triangle CPD$ (By SSS Congruence Rule).
1. $OA = PC$ (Radii of congruent circles)
2. $OB = PD$ (Radii of congruent circles)
3. $AB = CD$ (Given)
$\therefore \triangle AOB \cong \triangle CPD$ (By SSS Congruence Rule).
By CPCT (Corresponding Parts of Congruent Triangles):
$\angle AOB = \angle CPD$.
$\angle AOB = \angle CPD$.
Proved.
Q2: Converse of Q1
Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are
equal.
Given: Two congruent circles with centres $O$ and $P$.
$\angle AOB = \angle CPD$.
$\angle AOB = \angle CPD$.
To Prove: Chord $AB = CD$.
Proof: In $\triangle AOB$ and $\triangle CPD$:
1. $OA = PC$ (Radii)
2. $\angle AOB = \angle CPD$ (Given)
3. $OB = PD$ (Radii)
$\therefore \triangle AOB \cong \triangle CPD$ (By SAS Congruence Rule).
1. $OA = PC$ (Radii)
2. $\angle AOB = \angle CPD$ (Given)
3. $OB = PD$ (Radii)
$\therefore \triangle AOB \cong \triangle CPD$ (By SAS Congruence Rule).
By CPCT:
$AB = CD$.
$AB = CD$.
Proved.