Circles
NCERT Solutions • Class 9 Maths • Chapter 9Exercise 9.1
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.
Given: Two congruent circles with centres $O$ and $P$.
Chords $AB$ and $CD$ are equal ($AB = CD$).
To Prove: $\angle AOB = \angle CPD$
Proof: In $\Delta AOB$ and $\Delta CPD$:
1. $OA = PC$ (Radii of congruent circles)
2. $OB = PD$ (Radii of congruent circles)
3. $AB = CD$ (Given)
$\therefore \Delta AOB \cong \Delta CPD$ (By SSS Rule).
Conclusion:
$\Rightarrow$ $\angle AOB = \angle CPD$ (By CPCT).
Chords $AB$ and $CD$ are equal ($AB = CD$).
To Prove: $\angle AOB = \angle CPD$
Proof: In $\Delta AOB$ and $\Delta CPD$:
1. $OA = PC$ (Radii of congruent circles)
2. $OB = PD$ (Radii of congruent circles)
3. $AB = CD$ (Given)
$\therefore \Delta AOB \cong \Delta CPD$ (By SSS Rule).
Conclusion:
$\Rightarrow$ $\angle AOB = \angle CPD$ (By CPCT).
2. 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 $P$.
Angles subtended by chords are equal ($\angle AOB = \angle CPD$).
To Prove: $AB = CD$
Proof: In $\Delta AOB$ and $\Delta CPD$:
1. $OA = PC$ (Radii of congruent circles)
2. $\angle AOB = \angle CPD$ (Given)
3. $OB = PD$ (Radii of congruent circles)
$\therefore \Delta AOB \cong \Delta CPD$ (By SAS Rule).
Conclusion:
$\Rightarrow$ $AB = CD$ (By CPCT).
Angles subtended by chords are equal ($\angle AOB = \angle CPD$).
To Prove: $AB = CD$
Proof: In $\Delta AOB$ and $\Delta CPD$:
1. $OA = PC$ (Radii of congruent circles)
2. $\angle AOB = \angle CPD$ (Given)
3. $OB = PD$ (Radii of congruent circles)
$\therefore \Delta AOB \cong \Delta CPD$ (By SAS Rule).
Conclusion:
$\Rightarrow$ $AB = CD$ (By CPCT).