`A B C D` is a trapezium in which `A B||D C` , `M\ a n d\ N` are the mid-points of `A D\ a n d\ B C` respectively. If `A B=12 C M ,\ M N=14 C M ,\ ` then `C D=`

Options

A. 10cm
B. 12cm
C. 14cm
D. 16cm

Answer

To solve the problem, we need to find the length of \( CD \) in trapezium \( ABCD \) where \( AB \parallel DC \), and \( M \) and \( N \) are the midpoints of \( AD \) and \( BC \) respectively. We are given that \( AB = 12 \, cm \) and \( MN = 14 \, cm \).

### Step-by-Step Solution:

1. **Understand the Properties of the Trapezium**:
In a trapezium, the line segment joining the midpoints of the non-parallel sides (in this case, \( AD \) and \( BC \)) is parallel to the two bases and its length is the average of the lengths of the two bases.

2. **Write the Formula for the Length of \( MN \)**:
The length of \( MN \) can be expressed using the formula:
\[ MN = \frac{AB + CD}{2} \]

3. **Substitute the Known Values**:
We know that:
- \( AB = 12 \, cm \)
- \( MN = 14 \, cm \)

Substitute these values into the formula:
\[ 14 = \frac{12 + CD}{2} \]

4. **Eliminate the Fraction**:
To eliminate the fraction, multiply both sides of the equation by 2:
\[ 28 = 12 + CD \]

5. **Solve for \( CD \)**:
Now, isolate \( CD \) by subtracting 12 from both sides:
\[ CD = 28 - 12 \]
\[ CD = 16 \, cm \]

Thus, the length of \( CD \) is \( 16 \, cm \).

### Final Answer:
\( CD = 16 \, cm \)