Class IX, CIRCLES (Exercise 10.2) | NCERT (CBSE) Mathematics

Chapter 10, Circles

   CBSE Class IX Mathematics
NCERT Solutions of Math Textbook Exercise 10.2

(page 173)

Q 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 centers.

Ans 1:

A 'Circle' is an array of points which are equidistant from a fixed point. This fixed point is called as the 'Centre' of the circle and this equal distance is the 'Radius' of the circle. Therefore, if we try to superimpose two circles of equal radius, then both circles will cover each other. Therefore, two circles can be congruent only if, they have equal radii.

Consider two congruent circles having centre O and O' and two chords AB and CD of equal lengths.

In ΔAOB and ΔCO'D, we have -

AB = CD (Chords of same length)

OA = O'C (Radii of congruent circles)

OB = O'D (Radii of congruent circles)

Hence, ΔAOB ≅ ΔCO'D (by SSS congruence rule)

∠AOB = ∠CO'D (By CPCT)

Hence, it is proved that equal chords of congruent circles subtend equal angles at their centers.

Q 2: Prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.

Ans 2:

Say, there are two congruent circles (circles of same radius) with centers as O and O' as shown in the following figure

In ΔAOB and ΔCO'D, we have -

∠AOB = ∠CO'D (Given)

OA = O'C (Radii of congruent circles)

OB = O'D (Radii of congruent circles)

Hence, ΔAOB ≅ ΔCO'D (by SSS congruence rule)

AB = CD (By CPCT)

Hence this is proved that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.