In the quadrilateral ABCD, if AB // CD, E is mid-point of side AD and F is mid-point of BC. If AB = 20 cm and EF = 16 cm, the length of side DC is :

1. 18 cm

2. 12 cm

3. 24 cm

4. 32 cm

Join BD. Let BD intersect EF at point O.

We know that,

In trapezium the line joining the mid-points of non-parallel sides are parallel to the parallel sides of trapezium.

∴ AB || EF || DC.

By mid-point theorem,

The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it.

By converse of mid-point theorem,

The straight line drawn through the mid-point of one side of a triangle parallel to another, bisects the third side.

Given,

⇒ EF || AB

⇒ EO || AB

In △ ABD,

E is mid-point of AD and EO || AB.

∴ O is mid-point of BD. (By converse of mid-point theorem)

∴ EO = $\dfrac{1}{2}AB$ (By mid-point theorem) ……….(1)

Given,

⇒ EF || DC

⇒ OF || DC

In △ BCD,

O is mid-point of BD and F is mid-point of BC.

∴ OF = $\dfrac{1}{2}CD$ (By mid-point theorem) ……….(2)

Adding equations (1) and (2), we get :

⇒ EO + OF = $\dfrac{1}{2}AB + \dfrac{1}{2}CD$

⇒ EF = $\dfrac{1}{2}(AB + CD)$

Substituting values we get :

$\Rightarrow 16 = \dfrac{1}{2}(20 + CD) \\[1em] \Rightarrow 16 \times 2 = 20 + CD \\[1em] \Rightarrow 32 = 20 + CD \\[1em] \Rightarrow CD = 32 - 20 = 12\text{ cm}.$

Hence, Option 2 is the correct option.