Q60 of 62 Page 10

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.

Given: PQ is a chord in a circle with center O and MN is a tangent drawn at point R on the circle PQ is parallel to MN

To Prove: R bisects the arc PRQ i.e. arc PR = arc PQ


Proof:


∠1 = ∠2 [Alternate Interior angles]


∠1 = ∠3 [angle between tangent and chord is equal to angle made by chord in alternate segment]


So, we have


∠2 = ∠3


QR = PR [Sides opposite to equal angles are equal]


As the equal chords cuts equal arcs in a circle.


Arc PR = arc RQ


R bisects the arc PRQ.


Hence Proved


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