Question

ABCD is a parallelogram and X, Y are the mid-points of sides AB and Cd respectively. Then quadrilateral AXCY is a :

• A. Parallelogram
• B. Rhombus
• C. Square
• D. Rectangle

Answer 1

The correct option is A Parallelogram

$DY=YC$

$AX=XB$

$DCBA$ is a Parallelogram

So, $AD=BC$..........(1)

Now, $CD=AB$

$\frac{1}{2}CD=\frac{1}{2}AB$

So, $BX=DY$..............(2)

$\angle O+\angle C={180}^0\Rightarrow \angle D={180}^0-\angle C$

$\angle B+\angle C={180}^0\Rightarrow \angle B={180}^0-\angle C$

So, $\angle D=\angle B$...............(3)

From eq (1), (2) and (3)

$\Delta ADY\cong \Delta XBC$ (SAS)

So, $AY=CX$ (C.P.C.T)

and $YC=AX$ (From eq (2))

So, $AXCY$ is a parallelogram

Option (A) is a correct.