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A chord PQ of a circle …

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A chord PQ of a circle is parallel to the tangent drawn at a point R on the circle.prove that R bisects the arc PRQ.

Posted by Neha Singla 8 years, 10 months ago

CBSE > Class 10 > Mathematics

1 answers

Rahul Sharma 7 years, 10 months ago

Given: Circle with centre O. PQ is the chord parallel to the tangent m at R

To prove: The point R bisects the arc PRQ.

Construction: Join OR intersecting PQ at S.

Proof:

OR ⊥<i>m </i>(Radius is perpendicular to the tangent at the point of contact)

PQ||<i>m</i> (given)

∴∠OSP = ∠OSQ = 90° (pair of corresponding angles)

In ΔOPS and ΔOQS

OP = OQ (Radius)

OS = OS (Common)

∠OSP = ∠OSQ

So,ΔOPS ≅ ΔOQS (RHS criterion)

⇒ ∠POS = ∠QOS (By C.P.C.T)

⇒ arc (PR) = arc (QR) (Angle subtended by the arc at the centre and measure of the arc is same)

∴ R bisect the arc PRQ.

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