Question

✨ Hi Friends ✨

Q)Explain equal chords of a circle subtend equal angles at the center.

Answer 1

if two chords are equal the angle subtended by them at centre are equal.this can be proved with the help of congruency.first prove them congruent by SSS then by cpct central angle will be equal

Answer 2

Given : In a circle C(O,r), chord AB = chord CD.

To Prove : ∠AOB = ∠COD.

Proof : In△AOB and △COD

AO = CO [radii of same circle]

BO = DO [radii of same circle]

Chord AB = Chord CD [given]

⇒ △AOB ≅ △COD [by SSS congruence axiom]

⇒ ∠AOB = ∠COD. [c.p.c.t.]

A circle is a collection of points whose every every point is equidistant from the centre. Thus, two circles can only be congruent when they the distance of every point of the both circle is equal from the centre.

Given,
AB = CD (Equal chords)
To prove,
∠AOB = ∠COD
Proof,
In ΔAOB and ΔCOD,
OA = OC (Radii)
OB = OD (Radii)
AB = CD (Given)
∴ ΔAOB ≅ ΔCOD (SSS congruence condition)
Thus, ∠AOB = ∠COD by CPCT.
Equal chords of congruent circles subtend equal angles at their centres.