If E, F, G, H ar respectively the mid points of the sides of a parallelogram ABCD, show that ar(EFGH)= 1/2 ar(ABCD)

Dear Student

If the question is “If P, Q, R and S are respectively the mid-points of the sides of a parallelogram ABCD taken in order then show that area (PQRS) is half of area (ABCD”, then the answer to this question is:

ABCD is a parallelogram

∴ AB = CD and AB||CD

⇒ ½ AB = ½ CD and AP||DR

⇒AP = DR and AP||DR

Thus, APRD is a parallelogram

ΔSPR and parallelogram APRD lie on the same base PR and between the same parallels PR and AD.

∴ area (ΔSPR) = ½ area (APRD) … (1)

Similarly, it can be proved that

area (ΔPQR) = ½area (PBCR) … (2)

Adding both sides of equations (1) and (2)

area (ΔSPR) + area (ΔPQR) = ½ [area (APRD) + area (PBCR)]

⇒ area (PQRS) = ½ area (ABCD)

Cheers!

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