Question

Question 8
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 to tangents at R.
To prove: R bisects the arc PRQ

Proof: $\angle 1=\angle 2$ [alternate interior angles]
$\angle 1=\angle 3$
[Angle between tangent and chord is equal to angle made by chord in alternate segment]
$\therefore \angle 2=\angle 3$
$\Rightarrow$ PR = QR [sides opposite to equal angles are equal]
So, R bisects PQ.