Question

ABCD is a trapezium in which side AB is parallel to DC and E is the mid point of side AD. If F is the mid point of BC,such that line segments EF is parallel to DC,then prove that EF=1/2(AB+DC)

Answer 1

see the image for the answer

Answer 2

Join AC and let P be the point of intersection of AC and LM.

 

In ΔADC, L is the mid point of AD and LP||DC. (Given)

 

∴ P is the mid point of AC.(Converse of Mid Point Theorem)

 

We know that, line joining the midpoint of any two sides of triangle is parallel to the third side and half of it.

:-
$lp = \frac{1}{2} dc \: \: \: \: \: \: \: \: (1)$
In ΔABC, M and P are the mid points of side BC and AC respectively,

 
$pm = \frac{1}{2} ab......(2)$
Adding (1) and (2), we get

$lp + pm = \frac{1}{2} ab + \frac{1}{2} dc$
$= lm = \frac{1}{2} (ab + dc)$