Question

In quadrilateral ABCD of the given figure, X and
Y are points on diagonal AC such that AX = CY and
BXDY is a parallelogram. Show that ABCD is a parallelogram. ​

Answer 1

Step-by-step explanation:

We have, BXDY is a parallelogram. Therefore the diagonals bisect each other

=>BO=DO, XO=YO............. (1)

Given, AX=CY.

Adding xo on both sides, we get,

AX+XO=CY+YO.......... from 1 we have XO=YO

AO=CO.

Since AO=CO, BO=DO, therefore the diagonals bisect each other

Or, ABCD is a parallelogram

Answer 2

Answer:

BXDY is a parallelogram, BD and XY are diagonals of parallelogram.

We know that diagonals of parallelogram bisect each other.

XO=YO…(i)

DO=BO…(ii)

AX=CY…(iii)

Adding (i) and (iii), we have

XO+AX=YO+CY

⇒  AO=CO…(iv)

From (ii) and (iv), we have

AO=CO and DO=BO

This implies that AC and BD bisect each other but AC and BD are diagonals of quadrilateral ABCD hence, ABCD is a parallelogram.