# E is the mid-point of the side AD of the trapezium ABCD with AB || DC. A line through E drawn parallel to AB intersect BC at F. Show that F is the mid-point of BC. [Hint: Join AC]

**Solution:**

Given, ABCD is a __trapezium__

E is the midpoint of the side AD with AB __parallel__ to DC.

A line through E drawn parallel to AB intersect BC at F.

We have to show that F is the midpoint of BC

The __midpoint theorem__ states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

Considering __triangle__ ADC,

E is the midpoint of AD

So, OE is parallel to DC

i.e.,OE || DC

By midpoint theorem,

O is the midpoint of AC

Considering triangle CBA,

O is the midpoint of AC

So, OF is parallel to AB

i.e.,OF || AB

By midpoint theorem,

F is the midpoint of BC

Therefore, it is shown that F is the midpoint of BC.

**✦ Try This: **In parallelogram ABCD, two point P and Q are taken on diagonal BD such that DP=BQ. Show that AP = CQ

**☛ Also Check:** NCERT Solutions for Class 9 Maths Chapter 8

**NCERT Exemplar Class 9 Maths Exercise 8.3 Problem 6**

## E is the mid-point of the side AD of the trapezium ABCD with AB || DC. A line through E drawn parallel to AB intersect BC at F. Show that F is the mid-point of BC. [Hint: Join AC]

**Summary:**

E is the mid-point of the side AD of the trapezium ABCD with AB || DC. A line through E drawn parallel to AB intersect BC at F. It is shown that F is the mid-point of BC by Midpoint theorem

**☛ Related Questions:**

- Through A, B and C, lines RQ, PR and QP have been drawn, respectively parallel to sides BC, CA and A . . . .
- D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle AB . . . .
- Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD suc . . . .

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