Question

If 𝐄, 𝐅, 𝐆 and 𝐇 are respectively the midpoints of the sides 𝐀𝐁, 𝐁𝐂, 𝐂𝐃 and 𝐀𝐃 of a parallelogram 𝐀𝐁𝐂𝐃, show that 𝐚𝐫(𝐄𝐅𝐆𝐇) = 𝟏/𝟐 𝐚𝐫(𝐀𝐁𝐂𝐃).

Answer 1

ABCD a parallelogram in which E,F,G,H are mid points of AB,BC,CD,AD
Construction: Draw HF parallel to AB and CD
AB is parallel and equal to HF .
Therefore, ABFH is a parallelogram
Since, △EFH and parallelogram ABFH lies on the same base HF and between same parallels AB and HF
∴ar(ΔEFH)=1/2 ​ar(ABHF)...............(1)
Now, DC is parallel and equal to HF.
Therefore, DCFH is a parallelogram.
Since, ΔGFH and parallelogram DCFH lies on the same base
HF and between same parallels DC and HF.
∴ar(ΔGFH)=1/2 ​ar(DCHF)...................(2)
From (1) and (2) we get,
ar(ΔEFH)+ar(ΔGFH)=1/2 ​ar(ABFH) + ½ ​ar(DCFH)
⇒ ar(EFGH) = ½ ​(ar(ABFH) + ar(DCFH))
⇒ ar(EFGH)= ½ ​ar(ABCD)