Question

In the figure below, O is the centre of the circle and AB, CD are chords, with ∠OAB = ∠OCD,

Prove that AB = CD.

Answer 1

Given: Chords AB and CD form angles with centre O of the circle such that ∠OAB = ∠OCD.

To prove: AB = CD

Construction: Draw OF ⊥ AB and OE ⊥ CD.

Proof:

We know that perpendicular from the centre of the circle to the chord bisects the chord.

⇒ AF = FB and CE = ED

In ΔOAF and ΔOCE:

∠OAF = ∠OCE (Given)

∠AFO = ∠CEO = 90° (By construction)

OA = OC (Radii of the same circle)

∴ ΔOAF ≅ ΔOCE (By angle angle side criterion)

⇒ AF = CE (By c.p.c.t.)

Now, AB = AF + FB

= 2AF

= 2CE

= CD (CE = )

Thus, AB = C.