Question

ABCD is a parallelogram x and y are the midpoints of the sides BC and CD respectively prove that area of a x y=
3 /8 area of parallelogram ABCd

Answer 1

In∠BCDIn∠BCD

X and Y are midpoint of sides BC and CD

XY||BD and XY=1/2BD

ar(∠CXY)=14⋅ar(∠ABC)ar(∠CXY)=14⋅ar(∠ABC)

ar(∠CYX)=18⋅ar(parallelogramABCD)−(1)ar(∠CYX)=18⋅ar(parallelogramABCD)-(1)

ParallelogramABCD and ∠ABX∠ABX

AD||BX

BX=12BCBX=12BC

∠ABX=14(∣∣gramABCD)−(2)∠ABX=14(∣∣gramABCD)-(2)

ar(∠AYD)=14ar(∣∣gramABCD)−(3)ar(∠AYD)=14ar(∣∣gramABCD)-(3)

ar(∣∣gramABCD)=ar(∠ABX)+ar(∠AYB)+ar(CYX)+ar(∠AXY)ar(∣∣gramABCD)=ar(∠ABX)+ar(∠AYB)+ar(CYX)+ar(∠AXY)

ar(AXY)=ar(∣∣gramABCD)−[ar(∠ABX)+a