Question

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.

Answer 1

Let EF intersect DB at G.

By converse of mid-point theorem, we know that a line drawn through the mid-point of any side of a triangle and parallel to another side, bisects the third side.

In ∆ABD, 

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

Therefore, G will be the mid-point of DB. 

As EF || AB and AB || CD, 

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

In ∆BCD, 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.