Question

If the line joining the mid points D and E of the sides AB and CA respectively is extended to point F such that
DE = EF, then which of the following statement(s) is/are correct?

• A. DFCB is a parallelogram
• B. DF = EC
• C. EF is half of BC
• D. AD = BC

Answer 1

Answer: (A), (C)

The correct options are
A DFCB is a parallelogram
C EF is half of BC


In the figure given,
D and E are the mid points of the sides AB and CA respectively.

We know that the line segment joining the mid points of the two sides of a triangle is parallel to the third side and its length is equal to half the measure of the third side.

$\text{So,}DE=\frac{BC}{2}$
also, DE = EF
$\text{or,}EF=\frac{BC}{2}$

$△ADE\cong △CFE\text{by SAS criteria}$
as, DE = FE (given)
$\angle DEA=\angle FEC\text{(vertically opposite angles})$
EA = EC (E is the mid point of AC)
$\therefore AD=CF=DB$
as, D is the mid point of AB

We know that the pair of opposite sides of a parallelogram are equal and parallel.
So, DF || BC
and, BC = 2$\times$DE
or, BC = DE + EF = DF
Hence, DFCB is a parallelogram.