source: ncert_exercises
topic: geometry
class_level: class_9
chapter: ch9
exercise: ex9.1
difficulty: intermediate
Class 9 | Ch9: Circles | Exercise 9.1
Key Theorems: T1: Equal chords of a circle subtend equal angles at the centre. T2: If angles subtended by chords at centre are equal, chords are equal (converse).
Q1. Recall that two circles are congruent if they have same radii. Prove that equal chords of congruent circles subtend equal angles at their centres. Answer: Let circles with centres O and O' have equal radii r. Chords AB=CD (equal chords). In △AOB and △CO'D: OA=O'C=r, OB=O'D=r, AB=CD (given). ∴ △AOB≅△CO'D by SSS. ∴ ∠AOB=∠CO'D by CPCT. ✅ Equal chords subtend equal angles.
Q2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal. Answer: Let ∠AOB=∠CO'D. OA=O'C=r, OB=O'D=r (equal radii). In △AOB and △CO'D: OA=O'C, OB=O'D, ∠AOB=∠CO'D. ∴ △AOB≅△CO'D by SAS. ∴ AB=CD by CPCT. ✅ Equal angles → equal chords.