$P$ and $Q$ are points on oppostie sides $A D$ and $B C$ of a parallelogram $A B C D$ such that $P Q$ passes through the point of intersection $O$ of its diagonals $A C$ and $B D$ . $P Q$ is bisected at O.

True

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False

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Solution

The correct option is A True

$A B C D$ is a parallelogram whose diagonals bisect each other at $O .$

In $△ O D P$ and $△ O B Q,$

$\Rightarrow$ $∠ B O Q = ∠ P O D$ [ Vertically opposite angles are equal ]

$\Rightarrow$ $∠ O B Q = ∠ O D P$ [ Alternate interior angles ]

$\Rightarrow$ $O B = O D$ [ Given ]

$∴$ $△ O D P ≅ △ O B Q$ [ By $A S A$ congruence rule ]

$∴$ $O P = O Q$ [ By C.P.C.T ]

Hence, $P Q$ is bisected at $O .$

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Thinking Process

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