ABCD is a trapezium in which AB || 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.

To Prove: F is the mid-point of BC.

Given: ABCD is a trapezium with AB||DC, BD is a diagonal and E is the midpoint of AD.

Concept Used:

Converse Midpoint Theorem: A line drawn through the mid-point of any side of a triangle and parallel to another side, bisects the third side.

Diagram:

Proof:

Let EF intersect DB at G

In ΔABD,

EF || AB and E is the mid-point of AD

Therefore,

G will be the mid-point of DB

EF || AB and

AB || CD

EF || CD (Two lines parallel to the same line are parallel to each other)

In ΔBCD,

GF || CD and

G is the mid-point of line BD. Therefore, by using converse of mid-point theorem, F is the mid-point of BC.

Hence, Proved.