M and N are points on opposite sides AD and BC of a parallelogram ABCD such that MN passes through the point of intersection O of its diagonals AC and BD. Show that MN is bisected at O.

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Solution

Given: A parallelogram ABCD

To prove: MN is bisected at O

Proof:

In $∆$ OAM and $∆$ OCN,

OA = OC (Diagonals of parallelogram bisect each other)

∠AOM = ∠CON (Vertically opposite angles)

∠MAO = ∠OCN (Alternate interior angles)

$∴$ By ASA congruence criteria,

$∆ O A M ≅ ∆ O C N$

$\Rightarrow O M = O N$ (CPCT)

Hence, MN is bisected at O.

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Similar questions

M and N are points on opposite sides AD and BC of a parallelogram ABCD such that MN passes through the point of intersection O of its diagonals AC and BD. Show that MN is bisected at O.

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.

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.

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 ___.

In a parallelogram ABCD, points M and N have been taken on opposite sides AB and CD respectively such that AM = CN. Show that AC and MN bisect each other.

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