NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.2 – Part 3

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Converse of mid-point theorem

Converse of Mid-Point Theorem

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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.

Parallelogram ABCD

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:

Quadrilateral is ABCD

Given, ABCD be a quadrilateral and P, Q, R and S are the mid points of AB, BC, CD and DA respectively.

Now, In,

  • R and S are the mid points of CD and DA respectively.

  • .

Similarly we can show that,

Thus, PQRS is parallelogram and PR and QS are the diagonals of the parallelogram PQRS. So, they will bisect each other.