# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.2 – Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Converse of mid-point theorem: It states that in a triangle line drawn from the mid-point of the one side of triangle, parallel to the other side intersect the third side at its at mid-point.

Q-5 In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see Fig.) . Show that the line segments AF and EC trisect the diagonal BD.

Solution:

• Given in parallelogram ABCD, E and F are the mid-point of side AB and CD respectively.

Proof:

• ABCD is a parallelogram
• Therefore,
• Also,

Now,

• (Opposite sides of parallelogram ABCD)
• (E and F are midpoints of side AB and CD)
• AECF is a parallelogram (AE and CF are parallel and equal to each other)
• (Opposite sides of a parallelogram)

Now, In

• F is midpoint of side DC and (as ) .
• P is the mid-point of DQ (Converse of mid-point theorem)
• … equation (1)

Similarly, In APB,

• E is midpoint of side AB and (as ) .
• Q is the mid-point of PB (Converse of mid-point theorem)
• … equation (2)

From equations (1) and (2) ,

• Hence, the line segments AF and EC trisect the diagonal BD.

Q-6 Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Solution: