ABCD is a parallelogram. X and Y are the mid-points of BC and CD respectively. Prove that ar(AXY) = 3/8 ar(ABCD)
Construction: Join BD.
Since X and Y are the mid points of sides BC and CD respectively, therefore in ∆BCD,
XY || BD and XY BD.
⇒ area (∆CYX) area (∆DBC)
[In ∆BCD if X is the mid point of BC and Y is the mid point of CD then, area (∆CYX) area (∆DBC)]
⇒ area (∆CYX) area (||gm ABCD)
[Area of parallelogram is twice the area of triangle made by the diagonal]
Since parallelogram ABCD and ∆ABX are between the same parallel lines AD and BC and BX BC.
∴ area (∆ABX) area (||gm ABCD)
Similarly, area (∆AYD) (||gm ABCD)
Now, area (∆AXY) = area (||gm ABCD) – [area (∆ABX) + area (∆AYD) + area (∆CYX)]