# 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. 8.30). Show that F is the mid-point of BC.

**Solution:**

Let EF intersect DB at G as shown below.

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 trapezium ABCD,

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

Therefore, G is the mid-point of DB. [Converse of mid-point theorem]

As 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 the converse of mid-point theorem, F is the mid-point of BC.

**Video Solution:**

## 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. 8.30). Show that F is the mid-point of BC

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.2 Question 4

**Summary:**

If ABCD is a trapezium in which AB || CD, BD is a diagonal and E is mid-point of AD, a line is drawn through E parallel to AB intersecting BC at F, then F is the mid-point of BC.

**ā Related Questions:**

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