Question

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

Answer 1

Solution :
Given, ABCD is a Parallelogram. M and N are points taken on the diagonal AC such that AM = CN.
Proof:-
We know that the diagonals of a Parallelogram bisect each other. Therefore Ac and BD bisect each other at O.
∴      OC = OA
But  CN = AM
∴      OC - CN = OA - AM
⇒     OM = ON
Thus, in quadrilateral BMDN diagonals BD and MN are such that OM = ON and OD = OB, i.e., the diagonal BD and MN bisect each other. 
Hence, BMDN is a parallelogram.

Answer 2

Answer:

Solution :Given, ABCD is a Parallelogram. M and N are points taken on the diagonal AC such that AM = CN.Proof:-We know that the diagonals of a Parallelogram bisect each other. Therefore Ac and BD bisect each other at O.∴      OC = OABut  CN = AM∴      OC - CN = OA - AM⇒     OM = ONThus, in quadrilateral BMDN diagonals BD and MN are such that OM = ON and OD = OB, i.e., the diagonal BD and MN bisect each other. Hence, BMDN is a parallelogram.