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The mid-point of the sides of a triangle along with any of the vertices as the fourth point makes a parallelogram. Then find the relation between area of $Δ ABC$ and area of the parallelogram so obtained.

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a r A F D E = 14 a r Δ A B C

a r A F D E = 12 a r Δ A B C

a r A F D E = 13 a r Δ A B C

a r A F D E = a r Δ A B C

answer is B.

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Detailed Solution

Given, on joining midpoints of a triangle along with a vertex as fourth point makes a parallelogram.
Question Image

Let D, E and F respectively be the mid points of the sides BC, CA and AB.
Then, all four triangles have equal area.
⇒

a r Δ A F E = a r Δ B F D = a r Δ E D C = a r Δ D E F

⇒ a r Δ D E F = 14 a r Δ A B C

Similarly,

a r Δ A F E = 14 a r Δ A B C

a r A F D E = a r Δ A F E + a r Δ D E F a r A F D E = 2 14 a r Δ A B C [Proved above] a r A F D E = 12 a r Δ A B C

Hence, the correct option is 2.

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