Question

Prove 14 ..........
.........

Answer 1

$here \: is \: the \: answer \: of \: your \: question$

Given :- M and N are the midpoints of parallelogram ABCD. AN and CM are joined & P and Q are the midpoints on AN and MC which are joined as PM and NQ respectively.

To prove :- PMQN is a parallelogram.

Construction :- Extend PM and joined it with D & NQ to B as shown in figure.

Proof :- ABCD is a parallelogram and M and N are the midpoints of AB and CD.

As ABCD I a parallelogram and M is the mid point of AB.

So, AM = MB = AB/2 .............(1)

Now, N is also a mid point of side CD of parallelogram ABCD.

So, DN = NC = DC/2 ..............(2)

As, ABCD is a parallelogram then all of it's sides are equal and parallel to each other.

Means,
AB = CD and AD = BC

AB ll CD and AD ll BC

As, AB = CD.

So, AM = NC [ from (1) and (2) ]

Now, we have AM ll NC and AB ll CD.

In quadrilateral ANCM we have,

AM ll NC and AM ll NC [ we proved that above ]

MB = DN and MB ll DN

As, DMBN is paparalleogram

So, AN ll MC and AM ll NC.

then, ½ AN ll ½ MC

mean, PN ll MQ ...........(3)

Also in DNBM parallelogram.

DN ll MB and DM ll NB.

So, PM ll NQ ..............(4)

Now, in quadrilateral PMNQ,

we have,

PN ll MQ [ from (3) ]

and PM ll NQ [ from (4) ]

So that's mean, PMNQ is also a parallelogram.

Hence, proved.