The area of a trapezium is `405 cm^(2)`. Its parallel sides are in the ratio 4 : 5 and the distance between them is 18 cm. Find the length of each of the parallel sides.

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and apply the formula for the area of a trapezium.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Area of the trapezium (A) = 405 cm²
- Distance between the parallel sides (height, h) = 18 cm
- Ratio of the parallel sides = 4 : 5

2. **Assume the Lengths of the Parallel Sides:**
- Let the lengths of the parallel sides be 4x cm and 5x cm, where x is a common multiplier.

3. **Use the Formula for the Area of a Trapezium:**
- The formula for the area of a trapezium is:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
- Here, \( b_1 = 4x \) and \( b_2 = 5x \), and \( h = 18 \) cm.
- Substitute the values into the formula:
\[ 405 = \frac{1}{2} \times (4x + 5x) \times 18 \]

4. **Simplify the Equation:**
- Combine the terms inside the parentheses:
\[ 405 = \frac{1}{2} \times (9x) \times 18 \]
- Simplify further:
\[ 405 = \frac{1}{2} \times 162x \]
- Multiply both sides by 2 to eliminate the fraction:
\[ 810 = 162x \]

5. **Solve for x:**
- Divide both sides by 162:
\[ x = \frac{810}{162} = 5 \]

6. **Find the Lengths of the Parallel Sides:**
- Now substitute the value of x back to find the lengths of the parallel sides:
- First side (4x):
\[ 4x = 4 \times 5 = 20 \text{ cm} \]
- Second side (5x):
\[ 5x = 5 \times 5 = 25 \text{ cm} \]

7. **Conclusion:**
- The lengths of the parallel sides are:
- First parallel side = 20 cm
- Second parallel side = 25 cm

### Final Answer:
- The lengths of the parallel sides are 20 cm and 25 cm.

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