ABCD is a trapezium, such that AB = CD and AD || BC. AD = 5cm, BC = 9cm. If area of ABCD is 35 sq.cm then CD is :

Options

A. `sqrt(29)` cm
B. 5cm
C. 6 cm
D. `sqrt(21)` cm

Answer

To solve the problem, we need to find the length of side CD in trapezium ABCD where AB = CD, AD || BC, AD = 5 cm, BC = 9 cm, and the area of trapezium ABCD is 35 sq. cm.

### Step-by-Step Solution:

1. **Identify the properties of the trapezium**:
- Since AD is parallel to BC, we can use the formula for the area of a trapezium:
\[ \text{Area} = \frac{1}{2} \times (AB + CD) \times h \]
- Here, AB = CD (let's denote this common length as x), and we know the lengths of AD and BC.

2. **Set up the area formula**:
- The area of trapezium ABCD is given as 35 sq. cm. Therefore, we can write:
\[ 35 = \frac{1}{2} \times (x + x) \times h \]
- This simplifies to:
\[ 35 = x \times h \]

3. **Calculate the height (h)**:
- The height (h) can be found using the formula for the height of a trapezium when the lengths of the parallel sides are known. The height can be derived from the difference in lengths of the parallel sides and the lengths of the non-parallel sides.
- However, we can also use the trapezium area formula directly. We know that:
\[ h = \frac{2 \times \text{Area}}{AB + CD} \]
- Since AB = CD = x, we can substitute:
\[ h = \frac{2 \times 35}{x + x} = \frac{70}{2x} = \frac{35}{x} \]

4. **Substitute h back into the area equation**:
- We have:
\[ 35 = x \times \frac{35}{x} \]
- This confirms that our area equation holds true.

5. **Use the trapezium properties**:
- Since AD and BC are the non-parallel sides, we can use the trapezium property that states:
\[ BC - AD = h \]
- Therefore:
\[ 9 - 5 = h \implies h = 4 \text{ cm} \]

6. **Substitute h back to find x**:
- Now we can substitute h back into the area equation:
\[ 35 = x \times 4 \]
- Solving for x gives:
\[ x = \frac{35}{4} = 8.75 \text{ cm} \]

7. **Conclusion**:
- Since AB = CD, we find that:
\[ CD = 8.75 \text{ cm} \]

### Final Answer:
CD = 8.75 cm