Points $M$ and $N$ are taken on the diagonal $A C$ of a parallelogram $A B C D$ such that $A M = C N$ . Prove that $B M D N$ is a parallelogram.

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Solution

Diagonals of parallelogram bisect each other, therefore
$O C = O A$
But $C N = A M$
$∴ O C + O N = O A + O M$
$O M = O N$
Thus, in quadrilateral BMDN , its diagonals bisect each other , therefore BMDN is a parallelogram.

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Similar questions

Q. State true or false:

Points

M

and

N

are taken on the diagonal

A C

of a parallelogram

A B C D

such that

A M = C N

, then

B M D N

is a parallelogram.

Q. Points M and N are taken on the diagonal AC of a parallelogram ABCD such that AM = CN.
BMDN is a __________.

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$Prove that : (i) Δ D M C = Δ B N A a n d s o C M = A N (i i) Δ A M D ≅ Δ C N B a n d s o A M = C N (i i i) ANCM is a parallelogram.$

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Points M and N are taken on the diagonal AC of a parallelogram ABCD such that AM=CN. Prove that BMDN is a parallelogram.

Diagonals of parallelogram bisect each other, thereforeOC=OABut CN=AM∴OC+ON=OA+OMOM=ONThus, in quadrilateral BMDN , its diagonals bisect each other , therefore BMDN is a parallelogram.

Points $M$ and $N$ are taken on the diagonal $A C$ of a parallelogram $A B C D$ such that $A M = C N$ . Prove that $B M D N$ is a parallelogram.

Diagonals of parallelogram bisect each other, therefore
$O C = O A$
But $C N = A M$
$∴ O C + O N = O A + O M$
$O M = O N$
Thus, in quadrilateral BMDN , its diagonals bisect each other , therefore BMDN is a parallelogram.