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Q : 5    D, E and F are respectively the mid-points of the sides BC, CA and AB of a  \small \Delta ABC. Show that

           (i) BDEF is a parallelogram.

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We have a triangle \DeltaABC such that D, E and F are the midpoints of the sides BC, CA and AB respectively.

Now, in \DeltaABC, 
F and E are the midpoints of the side AB and AC.
Therefore according to mid-point theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half of the third side.
\therefore EF || BC or EF || BD

also, EF = 1/2 (BC)
\Rightarrow EF = BD [ D is the mid point of BC]
Similarly, ED || BF and ED = FB
Hence BDEF is a parallelogram.

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