- 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.
Solution:
Two congruent circles with centres O and O′. AB and CD are equal chords of the circles with centres O and O′ respectively.
To Prove : ∠AOB = ∠CO’D
Proof :
In triangles AOB and CO’D,
OA = O’C [since radii of congruent circles)
OB = O’D [Since radii of congruent circles]
AB = CD [Given]
⇒ ∆AOB ≅ ∆CO′D [SSS axiom]
⇒ ∠AOB ≅ ∠CO′D (CPCT)
- Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Solution:
Two congruent circles with centres O and O′. AB and CD are chords of circles with centre O and O′ respectively such that ∠AOB = ∠CO′D
To Prove : AB = CD
Proof : In triangles AOB and CO′D,
AO = CO[Radii of congruent circle]
BO = DO [Radii of congruent circle]
∠AOB = ∠CO′D [Given]
⇒ ∆AOB ≅ ∆CO′D [SAS axiom]
⇒ AB = CD [CPCT]
1 thought on “Circles – Exercise 10.2 – Class IX”
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