Question

$D,E$ and $F$ are respectively the mid-points of the sides $BC,CA$ and $AB$ of a $△ABC$
Show that $BDEF$ is a parallelogram.

Answer 1

In $△ABC$,

$\because F$ is the mid-point of side $AB$ and

$E$ is the mid-point of side $AC$

$\therefore EF\parallel BC$

$\because$ In a triangle, the line segment joining the mid-points of any two sides is parallel to the third side.

$\Rightarrow EF\parallel BD$ ......$\left(1\right)$

In $△ABC$,

$\because D$ is the mid-point of side $BC$ and

$E$ is the mid-point of side $AC$

$\therefore DE\parallel AB$

$\because$ In a triangle, the line segment joining the mid-points of any two sides is parallel to the third side.

$\Rightarrow DE\parallel FB$ ......$\left(2\right)$

Now, $FE\parallel BD$ and $DE\parallel FB$ from $\left(1\right)$ and $\left(2\right)$

In $BDEF$, both pairs of opposite sides are parallel

$\therefore BDEF$ is a parallelogram.