Question

E and F are respectively the midpoints of the non-parallel sides AD and BC of a trapezium ABCD.

• A. $EF||AB$
• B. $EF=\frac{1}{2}(AB+CD)$
• C. $DC=EB$
• D. None is true

Answer 1

Answer: (A), (B)

The correct options are
A

$EF||AB$

B

$EF=\frac{1}{2}(AB+CD)$

Given ABCD is a trapezium in which AB||CD. Also, E and F are respectively the mid-points of sides AD and BC.

In $\Delta GCB$, E and F are the mid-points of BG and BC respectively.

EF||GC [By midpoint theorem]

But GC||AB or CD||AB

$\therefore EF||AB$

In $\Delta ADB,AB||EO$ and E is the midpoint of AD.

Therefore by the converse of mid-point theorem, O is the midpoint of BD.
Also, $EO=\frac{1}{2}AB$

In $\Delta BDC,OF||CD$ and O is the mid-point of BD.
$\therefore OF=\frac{1}{2}CD$ [by converse of mid-point theorem]

Adding Eqs. (i) and (ii), we get

$EO+OF=\frac{1}{2}AB+\frac{1}{2}CD\Rightarrow EF=\frac{1}{2}(AB+CD)$