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# D, E and F are respectively the mid-points of the sides BC, CA and AB of a ∆ ABC. Show that (i) BDEF is a parallelogram.

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.

Views

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

Now, in $\Delta$ABC,
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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