Question

A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.

Answer 1

Given: Chord PQ is parallel tangent at R.

To prove: R bisects the arc PRQ.

Proof:

Since PQ || tangent at R.

∠1 = ∠2 [alternate interior angles]

∠1 = ∠3

[angle between tangent and chord is equal to angle made by chord in alternate segment]

So, ∠2 = ∠3

⇒ PR = QR [sides opposite to equal angles are equal]

Hence, clearly R bisects PQ.