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**Question 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. Solution: **We have a circle having its centre at O and two equal chords AB and CD such that they subtend âˆ AOB and âˆ COD respectively at the centre, i.e. at O.

We have to prove that âˆ AOB = âˆ COD

Now, in Î”AOB and Î”COD, we have

AO = CO [Radii of the same circle]

BO = DO [Radii of the same circle]

AB = CD [Given]

âˆ´ Î”AOB â‰Œ Î”COD [SSS criterion]

â‡’ Their corresponding parts are equal.

âˆ´ âˆ AOB = âˆ COD

**Question 2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal. Solution:** We have a circle having its centre at O, and its two chords AB and CD such that

âˆ AOB = âˆ COD

We have to prove that AB = CD

âˆµ In Î”AOB and Î”COD, we have:

AO = CO [Radii of the same circle]

BO = DO [Radii of the same circle]

âˆ AOB = âˆ COD [Given]

âˆ´ Î”AOB â‰Œ Î”COD [SAS criterion]

âˆ´ Their corresponding parts are equal, i.e. AB = CD

**PERPENDICULAR FROM THE CENTRE TO A CHORD**

**REMEMBER**

- The perpendicular from the centre of a circle to a chord bisects the chord.
- The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
- There is one and only one circle passing through three given non-collinear points.

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