P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Then, point O ___.

bisects PQ

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is a trisection point of PQ

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does not bisect PQ but only lies on PQ

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None of these

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Solution

The correct option is A
bisects PQ

Given: ABCD is a parallelogram whose diagonals bisect each other at O.

In $Δ O D P a n d Δ O B Q,$

$∠ B O Q = ∠ P O D$ [Vertically opposite angles]

$∠ O B Q = ∠ O D P$ [atternate interior angles]

and OB = OD [given]

$∴ Δ O D P ≅ Δ O B Q$ [by ASA congruence rule]

$∴ O P = O Q$ [CPCT]

So, PQ is bisected at O, or O bisects PQ.

Suggest Corrections

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Question 14
P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O is its diagonals AC and BD. Show that PQ is bisected at 0.

Thinking Process

Firstly, prove that $Δ O D P a n d Δ O B Q$ are congruent by ASA rule. Further show the required result by CPCT rule.