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Equal Chords of a Circle

PQ and RS are...

Question

PQ and RS are two parallel chords of a circle in the given figure. If lines RP and SQ(produced) intersect at O, then OP $=$ __________.

A

PR

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B

RS

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C

OQ

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D

QS

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Solution

The correct option is C OQ
In cyclic quadrilateral $P Q R S$ :
$∵ P Q | | R S \Rightarrow P R = Q S$
We know from the properties of circle, $O P \cdot O R = O Q \cdot O S$
Above can also be written as $O P \cdot (O P + P R) = O Q \cdot (O Q + Q S) \Rightarrow O P^{2} - O Q^{2} = Q S (O Q - O P) \Rightarrow (O P - O Q) (O P + O Q + Q S) = 0 ∴ O P = O Q$

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