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 : 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 $\angle 1$ with tangent MN and $\angle 2$ in alternates segment of circle so $\angle 1=\angle 2$

$MRN||PQ$

$\therefore \angle 1=\angle 3$ [Alternate interior angles]

$\Rightarrow \angle 2=\angle 3$

$\Rightarrow PR=RQ$ [Sides opp. to equal $\angle s$ in $\Delta RPQ$]

$\because$ Equal chords subtend equal arcs in a circle so

$arcPR=arcRQ$

or R bisect the $arcPRQ$. Hence proved.