Exercise 9.1 Solutions (Class 9)
Circles
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 O’. They have equal radii, so OA = OB = O’C = O’D. We are also given that the chords are equal, i.e., AB = CD.
To Prove: The angles subtended at the centres are equal, i.e., ∠AOB = ∠CO’D.
Proof:
Consider the two triangles ΔAOB and ΔCO’D.
OA = O'C(Radii of congruent circles are equal)OB = O'D(Radii of congruent circles are equal)AB = CD(Given that chords are equal)
By the Side-Side-Side (SSS) congruence rule, ΔAOB ≅ ΔCO’D.
Since the triangles are congruent, their corresponding parts are equal. Therefore, by CPCT (Corresponding Parts of Congruent Triangles):
∠AOB = ∠CO’D.
Hence, proved.
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 O’. They have equal radii (OA = OB = O’C = O’D). The angles subtended by the chords at the centres are equal, i.e., ∠AOB = ∠CO’D.
To Prove: The chords are equal, i.e., AB = CD.
Proof:
Consider the two triangles ΔAOB and ΔCO’D.
OA = O'C(Radii of congruent circles are equal)∠AOB = ∠CO'D(Given)OB = O'D(Radii of congruent circles are equal)
By the Side-Angle-Side (SAS) congruence rule, ΔAOB ≅ ΔCO’D.
Since the triangles are congruent, their corresponding parts are equal. Therefore, by CPCT (Corresponding Parts of Congruent Triangles):
AB = CD.
Hence, proved.
