In the adjoining figure, If $O$ is the center of the circle and the angle $x = y$ , then which of the following is true?

2 O A = R A

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O A = P A

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P Q = R S

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R O = R S

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Solution

The correct option is A $P Q = R S$
From the above figure $R S = R A + A S$ and $P Q = P A + Q A$
$Δ R D A$ and $Δ A D P$ are congruence triangle
Hence $R A = A P$
now in $Δ A Q M$ and $Δ A S M$ $∠ Q A M = ∠ S A M$
Hence $Δ A Q M$ and $Δ A S M$ are congruence triangle
Hence $A Q = A S$
Hence $R S = P Q$

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