Question

If E, F G and H are respectively themidpoints of the sides AB, BC, CD and ADof a parallelogram ABCD, show thatar(EFGH) 1ar(ABCD)2=.

Answer 1

Correct Question:-

If E,F,G and H are respectively the mid point of the sides of a parallelogram ABCD ,Show that ar(EFGH) = 1/2 ar(ABCD)

Given:-

E ,F ,G and H are the mid points of the side of a parallelogram ABCD, respectively.

To Prove:-

ar(EFGH) = 1/2 ar(ABCD)

$\bf\:Construction:-$

H and F are joined.

Proof:-

AD ∥ BC and AD = BC (opposite sides of ∥gm)

$\implies\:\pink{\underline{\boxed{\bf{\dfrac{1}{2}AD=\dfrac{1}{2}BC}}}}$

Also,

AH ∥ BF and DH = CF ( H and F are mid point)

∴ ABFH and HFCD are ∥gm

Now,

We know that ∆EFH and ∥gm ABFH both lies on the same FH the common base and in between the same parallel lines AB and HF.

∴ Area of ∆EFH = 1/2 area of ∥gm ABFH............(i)

and,

Area of ∆ GHF = 1/2 area of ∥gm HFCD...............(ii)

Adding both eq (i) and eq (ii)

Area of ∆ EFH + area of ∆GHF = 1/2 area of ∥gm ABHF + 1/2 area of ∥gm HFCD

area of ∥gm EFGH = area of ∥gm ABFH

$\green{\underline{\boxed{\bf{ar(EFGH)=\dfrac{1}{2}\:ar(ABCD)}}}}$