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)

  • 37

in parallelogran ABCD, EFGH are the midpoints so E,F,G,H are the midpoints on AB, BC. CD. DA respectively

EF =1/2 AC and is parallel to AC (midpoint theorm)

GH=1/2 AC and is parallel to AC(midpoint theorm)    so, EF=GH

similarly , EH and GF are 1/2 of  diagonal BD            EH=GF

since , EF=GH & EH=GF

and EFGH=1/2 ABCD since they are the midpoints od parallelogram ABCD

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