Class 9 Mathematics
Chapter 9: Circles
Exercise 9.1 – Stepwise Solutions
Important Concept Used
In congruent circles, radii are equal.
If in two triangles, three corresponding sides are equal, then the triangles are congruent by SSS Congruence Rule.
By CPCT, corresponding angles and sides of congruent triangles are equal.
Q1. 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 O’.
AB and CD are equal chords of the two congruent circles.
To prove:
∠AOB = ∠CO’D
Construction / Observation:
Join OA, OB, O’C and O’D.
Proof:
Since the circles are congruent, their radii are equal.
So,
OA = O’C
OB = O’D
Also given,
AB = CD
Therefore, in triangles AOB and CO’D:
OA = O’C
OB = O’D
AB = CD
Hence,
ΔAOB ≅ ΔCO’D by SSS congruence rule.
Therefore, by CPCT,
∠AOB = ∠CO’D
Hence proved that equal chords of congruent circles subtend equal angles at their centres.
Q2. 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 O’.
Chords AB and CD subtend equal angles at their centres.
That is,
∠AOB = ∠CO’D
To prove:
AB = CD
Construction / Observation:
Join OA, OB, O’C and O’D.
Proof:
Since the circles are congruent, their radii are equal.
So,
OA = O’C
OB = O’D
Also,
∠AOB = ∠CO’D
Therefore, in triangles AOB and CO’D:
OA = O’C
OB = O’D
∠AOB = ∠CO’D
Hence,
ΔAOB ≅ ΔCO’D by SAS congruence rule.
Therefore, by CPCT,
AB = CD
Hence proved that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.