Question

If E, F, G and H are respectively the mid-points of the sides of a parallelogram ABCD show that

ar (EFGH) ar (ABCD)

Answer 1

Let us join HF.

In parallelogram ABCD,

AD = BC and AD || BC (Opposite sides of a parallelogram are equal and parallel)

AB = CD (Opposite sides of a parallelogram are equal)

and AH || BF

⇒ AH = BF and AH || BF ( H and F are the mid-points of AD and BC)

Therefore, ABFH is a parallelogram.

Since ΔHEF and parallelogram ABFH are on the same base HF and between the same parallel lines AB and HF,

∴ Area (ΔHEF) = Area (ABFH) ... (1)

Similarly, it can be proved that

Area (ΔHGF) = Area (HDCF) ... (2)

On adding equations (1) and (2), we obtain