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Question
Points M and N are taken on the diagonal AC of a parallelogram ABCD such that AM = CN. Prove that BMDN is a parallelogram.
Solution 1
Points M are N taken on the diagonal AC of a parallelogram ABCD such that.
Prove that BMDN is a parallelogram
construction: Join B to D to meet AC in O.
Proof: We know that the diagonals of a parallelogram bisect each other.
Now, AC and BD bisect each other at O.
OC = OA
AM = CN
OA - AM = OC - CN
OM = ON
Thus in a quadrilateral BMDN, diagonal BD and MN are such that OM = ON and OD = OB
Therefore the diagonals AC and PQ bisect each other.
Hence BMDN is a parallelogram
Solution 2
Join BD.
The diagonals of a parallelogram bisect each other.
Therefore, AC and BD bisect each other.
⇒ OA = OC
But AM = CN
Therefore,OA - AM = OC - CN
⇒ OM = ON
Therefore, in quadrilateral BMDN,
OM = ON and OD = OB
⇒ Diagonals MN and BD bisect each other
⇒ BMDN is a parallelogram.
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