The difference between the lengths of the parallel sides of a trapezium is 8 cm, the perpendicular distance between these sides is 24 cm and the area of the trapezium is 312 `cm^(2)`. Find the length of each of the parallel sides.

Answer

To solve the problem, we need to find the lengths of the two parallel sides of the trapezium, given the following information:

1. The difference between the lengths of the parallel sides (L2 - L1) is 8 cm.
2. The perpendicular distance between the parallel sides (height, h) is 24 cm.
3. The area of the trapezium is 312 cm².

Let's denote:
- L1 = length of the shorter parallel side
- L2 = length of the longer parallel side

### Step 1: Set up the equations
From the information given, we can set up the following equations:

1. **Equation 1 (from the difference of lengths)**:
\[ L2 - L1 = 8 \quad \text{(1)} \]

2. **Equation 2 (from the area of the trapezium)**:
The area \( A \) of a trapezium is given by the formula:
\[ A = \frac{1}{2} \times (L1 + L2) \times h \]
Plugging in the values we have:
\[ 312 = \frac{1}{2} \times (L1 + L2) \times 24 \quad \text{(2)} \]

### Step 2: Simplify Equation 2
To simplify Equation 2, we can multiply both sides by 2 to eliminate the fraction:
\[ 624 = (L1 + L2) \times 24 \]
Now, divide both sides by 24:
\[ L1 + L2 = \frac{624}{24} = 26 \quad \text{(3)} \]

### Step 3: Solve the system of equations
Now we have two equations:
1. \( L2 - L1 = 8 \) (Equation 1)
2. \( L1 + L2 = 26 \) (Equation 3)

We can solve these equations simultaneously.

From Equation 1, we can express \( L2 \) in terms of \( L1 \):
\[ L2 = L1 + 8 \quad \text{(4)} \]

### Step 4: Substitute Equation 4 into Equation 3
Now substitute Equation 4 into Equation 3:
\[ L1 + (L1 + 8) = 26 \]
This simplifies to:
\[ 2L1 + 8 = 26 \]
Subtract 8 from both sides:
\[ 2L1 = 18 \]
Now divide by 2:
\[ L1 = 9 \quad \text{(5)} \]

### Step 5: Find \( L2 \)
Now substitute \( L1 \) back into Equation 4 to find \( L2 \):
\[ L2 = 9 + 8 = 17 \quad \text{(6)} \]

### Conclusion
The lengths of the parallel sides of the trapezium are:
- \( L1 = 9 \) cm
- \( L2 = 17 \) cm