Parallelogram :

A quadrilateral inwhich both pairs of opposite sides are parallel is called a parallelogram.

Aquadrilateral is a parallelogram if

i)Itsopposite sides are equal

ii)its opposite angles are equal

iii)diagonals bisect each other

iv)a pair of opposite sides is equal and parallel.

Converseof mid point theorem:

Theline drawn through the midpoint of one side of a triangle, parallel to anotherside bisect the third side.

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Given,

ABCD is a parallelogram. E and F are the mid-points of sides AB and CDrespectively.

To show: line segmentsAF and EC trisect the diagonal BD.

Proof,

ABCD is a parallelogram

Therefore, AB || CD

also, AE || FC

Now,

AB = CD

(Opposite sides of parallelogram ABCD)

1/2 AB = 1/2 CD

AE = FC

(E and F aremidpoints of side AB and CD)

Since a pair of opposite sides of aquadrilateral AECF is equal and parallel.

so,AECF is aparallelogram

Then, AF||EC,

AP||EQ & FP||CQ

(Since opposite sides of aparallelogram are parallel)

Now,

In ΔDQC,

F is mid point of side DC & FP || CQ

(as AF || EC).

So,P is themid-point of DQ

(by Converse of mid-point theorem)

DP = PQ — (i)

Similarly,

In APB,

E is mid point of side AB and EQ || AP

(as AF || EC).

So,Qis the mid-point of PB

(by Converse of mid-point theorem)

PQ = QB —(ii)

From equations (i) and (ii),

DP = PQ = BQ

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

tqsm