Question

7. 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.
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Answer 1

Given : I a circle a chord PQ and a tangent MRN at R such that QP∣∣MRN

To prove : R bisects the arc PRQ

Construction : Join RP and RQ

Proof : Chord RP subtends ∠1 with tangent MN and ∠2 in alternates segment of circle so ∠1=∠2

MRN∣∣PQ

∴∠1=∠3 [Alternate interior angles]

⇒∠2=∠3

⇒PR=RQ [Sides opp. to equal ∠s in ΔRPQ]

∵ Equal chords subtend equal arcs in a circle so

arcPR=arc RQ

or R bisect the arc PRQ. Hence proved.

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