Question

AB and AC are two equal chords of a circle with centre o such that LABO and LCBO are equal. Prove that AB = BC. 

Answer 1

Given: AB = AC, ∠ ABO = ∠ CBO 

To Prove: AB = BC 

Construction : Draw ON  ⊥ AB and OM ⊥ BC 

Proof : In triangles  BNO and BMO, 

∠ NBO = ∠ MBO   (Given)

∠ BNO = ∠ BMO   (Each 90 ° )

BO= BO        (common)

Thus , Δ BNO ≅ Δ BMO     (By AAS)

⇒ BN =BM 

⇒ 2 BN = 2 BM        (Since perpendicular drawn from the centre bisects the chord)

⇒ AB = BC

Hence Proved.