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| source: ncert_exercises |
| topic: geometry |
| class_level: class_9 |
| chapter: ch9 |
| exercise: ex9.1 |
| difficulty: intermediate |
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| Class 9 | Ch9: Circles | Exercise 9.1 |
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| 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). |
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| 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. |
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| 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. |
