in a parrarelogram abcd x and y are the middle points the sides ad and bc respectively if p be any point innxy prosuced prove the area of quadrilateral aecb is 2by 3 of the parrarelogram abcd
Answers
Step-by-step explanation:
X and Y are the mid-points of sides BC and CD.
In △BCD,
XY∥BD and XY=
2
1
BD {From the mid point theorem}
⇒ ar(△CYX)=
4
1
ar(△DBC) {property of triangle having mid points}
⇒ ar(△CYX)=
8
1
ar(∥
gm
ABCD) [ Area of parallelogram is twice the area of triangle made by diagona ] --- ( 1 )
Since, parallelogram ABCD and △ABX are between the same parallel lines AD and BC and BX=
2
1
BC.
⇒ ar(△ABX)=
4
1
ar(∥
gm
ABCD) ----- ( 2 )
Similarly, ar(△AYD)=
4
1
ar(∥
gm
ABCD) ----- ( 3 )
Now, ar(△AXY)=ar(∥
gm
ABCD)−[ar(△ABX)+ar(△AYD)+ar(△CYX)]
=ar(∥
gn
ABCD)−[
4
1
ar(∥
gm
ABCD+
4
1
ar(∥
gm
ABCD)+
8
1
ar(∥
gm
ABCD)] [ From ( 1 ) ( 2 ) and ( 3 ) ]
=ar(∥
gm
ABCD)−
8
5
(∥
gm
ABCD)
⇒ ar(△AXY)=
8
3
ar(∥
gm
ABCD)
∴ ar(∥
gm
ABCD)=
3
8
×ar(△AXY)