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Byju's Answer

Standard IX

Mathematics

Properties of Diagonal of a Parallelogram

M and N are p...

Question

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
(D iagonals of parallelogram bisect each other)
∠AOM = ∠CON (Vertically opposite angles)
∠MAO = ∠OCN (Alternate interior angles)
∴ By ASA congruence criteria,
∆OAM ≅ ∆OCN
⇒ OM = ON (CPCT)
Hence, MN is bisected at O.

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Similar questions

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.

Q. In trapezium ABCD,

M \in ¯ ¯¯¯¯¯¯¯ ¯ A D, n \in ¯ ¯¯¯¯¯¯ ¯ B C

are the points such that

\frac{A M}{M D} = \frac{B N}{N C} = \frac{2}{3}

.
The diagonal

¯ ¯¯¯¯¯¯ ¯ A C

intersects

¯ ¯¯¯¯¯¯¯¯¯ ¯ M N

O

. Then find the value of

\frac{A O}{A C}

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