Q. 84.3( 31 Votes )

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 :

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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