Question

ABCD is a parallelogram in which E and F are the midpoints of AB and CD respectively. prove that AECF is a parallelogram​

Answer 1

Parallelogram :

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

A quadrilateral is a parallelogram if

i) Its opposite sides are equal

ii) its opposite angles are equal

iii) its diagonals bisect each other

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

Converse of mid point theorem:

The line drawn through the midpoint of one side of a triangle, parallel to another side bisect the third side.

=======================

Given,

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

To show: line segments AF 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 are
midpoints of side AB and CD)

Since a pair of opposite sides of a quadrilateral AECF is equal and parallel.
so,AECF is a parallelogram.

Then, AF||EC,

AP||EQ & FP||CQ

(Since opposite sides of a

parallelogram are parallel)

Answer 2

Answer:

In ∆CDQ

DF = CF

DF//QC

DP = PQ -----(i) ( Converse repeating

theorem )

In ∆ABP

AE = EB ( Given )

EQ // AP ( Midpoint theorem )

PQ = QP -----(ii) ( Converse Midpoint

theorem )

From (i) and (ii)

PQ = QB = DP

Line Segments AF and EC bisect the diagonal BD