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)]

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