Question

$ABCD$ is a trapezium in which $AB\parallel DC,BD$ is a diagonal and $E$ is the mid-point of $AD$. A line is drawn through $E$ parallel to $AB$, intersecting $BC$ at $F$ (see Fig). Show that $F$ is the mid-point of $BC$.

Answer 1

In $△DAB$,

Let line $EF$ intersect side $DB$ at $G$, such that $E-G-F$ is a line.

$E$ is the midpoint of side $AD$ and $EG\parallel AB$.

By converse of midpoint theorem,

$G$ is the midpoint of side $BD$ .....(1)

$AB\parallel DC$ ....given

$AB\parallel EF$ ....given

$\therefore AB\parallel DC\parallel EF$ ....(2)

In $△BDC$,

$G$ is the midpoint of side $BD$ ...from (1)

and $DC\parallel GF$ ....from (2)

By converse of midpoint theorem,

$F$ is the midpoint of side $BC$