Converse of mid-point theorem: 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

Parallelogram ABCD

E and F are the mid-point of side AB and CD respectively.

To prove,

Proof,

ABCD is a parallelogram

Therefore, and

Now,

(Opposite sides of parallelogram ABCD)

(E and F are midpoints of side AB and CD)

Therefore, 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 ).

Therefore, 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:

Given,

ABCD is a quadrilateral

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively.

Now, in ,

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

Therefore by midpoint theorem, .

Similarly we can show that,

Thus, PQRS is parallelogram.

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