Question

ABCD is a parallelogram . X and Y are the midpoints of the opposite sides AB and CD respectively. Prove that AXCY is a parallelogram.

Answer 1

Given: X and Y are the mid-points of opposite sides AB and DC of a parallelogram ABCD.
AX = XB and DY = YC
To Prove: PXQY is a parallelogram.
Proof: AB = DC 
... AB =   DC 
XB = DY......(i)
Also AB ïï DC...... (opp. sides of a ïïgm)
      XB ïï DY.....(ii)
Since in quadrilateral XBYD, XB = DY and XB ïï DY
... XBYD is a parallelogram.
... DX ïï YB ⇒ PX ïï YQ......(iii)
Similarly we can prove, PY ïï XQ ......(iv)
From (iii) and (iv), we get PXQY is a parallelogram. 

Answer 2

Refer the attachment.

ABCD is a ||gm


=> AB || CD

and AB = CD

(since opposite sides are equal and parallel in a parallelogram)

Now,

since AB = CD

=> 1/2 AB = 1/2 CD

=> AX = CY

(since X is the midpoint, AX is half of AB and Y is midpoint, CY is half of CD)

again,

since AB || CD

=> AX || CY

(Because they are the part of AB and CD)


So now we have,


AX = CY
AX || CY


since Opposite sides are equal and parallel, AXCY is a parallelogram.


Hope it helps dear friend ☺️✌️