Question

Question 4
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 the given figure). 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 $\Delta$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.
$\therefore$ EF || CD ( two lines parallel to the same line are parallel to each other)
In $\Delta$ BCD, GF || CD and G is the mid-point of line BD. Therefore, by using converse of midpoint theorem, F is the mid-point of BC.