The area of the trapezium is 105 `"cm"^2` and its height is 7 cm. If one of the parallel sides is longer than the other by 6 cm, find the length of parallel sides.

Answer

To solve the problem step by step, we will use the formula for the area of a trapezium and the given information.

### Step-by-Step Solution:

1. **Understand the given information**:
- Area of the trapezium (A) = 105 cm²
- Height (h) = 7 cm
- Let the lengths of the parallel sides be \( x \) and \( x + 6 \) (since one side is longer than the other by 6 cm).

2. **Use the formula for the area of a trapezium**:
The formula for the area of a trapezium is:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \( b_1 \) and \( b_2 \) are the lengths of the parallel sides.

3. **Substitute the known values into the formula**:
\[ 105 = \frac{1}{2} \times (x + (x + 6)) \times 7 \]

4. **Simplify the equation**:
\[ 105 = \frac{1}{2} \times (2x + 6) \times 7 \]
\[ 105 = \frac{7}{2} \times (2x + 6) \]

5. **Multiply both sides by 2 to eliminate the fraction**:
\[ 210 = 7 \times (2x + 6) \]

6. **Divide both sides by 7**:
\[ 30 = 2x + 6 \]

7. **Isolate \( 2x \)**:
\[ 2x = 30 - 6 \]
\[ 2x = 24 \]

8. **Solve for \( x \)**:
\[ x = \frac{24}{2} = 12 \]

9. **Find the lengths of the parallel sides**:
- First parallel side \( b_1 = x = 12 \) cm
- Second parallel side \( b_2 = x + 6 = 12 + 6 = 18 \) cm

### Final Answer:
The lengths of the parallel sides are:
- First parallel side = 12 cm
- Second parallel side = 18 cm