Solution

A circle is a collection 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 called as radius of the

circle. And thus, the shape of a circle depends on its radius. Therefore, it can be observed that if we try to superimpose two circles of equal radius, then both circles will cover each other. Therefore, two circles are congruent if they have equal radius. Consider two congruent circles having centre $O$ and $O^{\prime}$ and two chords $A B$ and $C D$ of equal lengths.

In $\triangle A O B$ and $\triangle C^{\prime} ’ D$,

$A B=C D$ (Chords of same length)

$OA=O^{\prime} C$ (Radii of congruent circles)

$OB=O$ ‘D (Radii of congruent circles)

$\triangle AOB \quad \triangle CO$ ‘D (SSS congruence rule) $\quad \therefore \cong \Rightarrow^{\circ}$

$AOB=\stackrel{\angle}{C} CO^{\prime} D$(By CPCT)

Hence, equal chords of congruent circles subtend equal angles at their centres.

Solution

Let us consider two congruent circles (circles of same radius) with centres as $O$ and $O^{\prime}$.

In $\triangle AOB$ and $\triangle CO^{\prime} D$,

$\angle A O B=\angle C^{\prime}$ D (Given)

$OA=O^{\prime} C$ (Radii of congruent circles)

$OB=O$ ‘D (Radii of congruent circles)

$\therefore \quad \triangle AOB \cong \triangle CO^{\prime} D$ (SSS congruence rule)

$\Rightarrow \quad AB=CD$ (By CPCT)

Hence, if chords of congruent circles subtend equal angles at their centres, then the chords are equal.