The parallel sides of a trapezium are in a ratio 2: 3 and their shortest distance is 12 cm. If the area of the trapezium is 480 sq. cm, the longer of the parallel sides is of length

Options

A. 36 cm
B. 42 cm
C. 48 cm
D. 56 cm

Answer

To find the length of the longer parallel side of the trapezium, we can follow these steps:

### Step 1: Define the lengths of the parallel sides
Let the lengths of the parallel sides be \(2x\) and \(3x\), based on the given ratio of 2:3.

### Step 2: Use the formula for the area of a trapezium
The area \(A\) of a trapezium can be calculated using the formula:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \(b_1\) and \(b_2\) are the lengths of the parallel sides, and \(h\) is the height. Here, we have:
\[ A = 480 \, \text{sq. cm}, \quad b_1 = 2x, \quad b_2 = 3x, \quad h = 12 \, \text{cm} \]

### Step 3: Substitute the values into the area formula
Substituting the values into the area formula gives:
\[ 480 = \frac{1}{2} \times (2x + 3x) \times 12 \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 480 = \frac{1}{2} \times 5x \times 12 \]
\[ 480 = 30x \]

### Step 5: Solve for \(x\)
To find \(x\), divide both sides by 30:
\[ x = \frac{480}{30} = 16 \]

### Step 6: Find the lengths of the parallel sides
Now, substitute \(x\) back to find the lengths of the parallel sides:
- The shorter side \(b_1 = 2x = 2 \times 16 = 32 \, \text{cm}\)
- The longer side \(b_2 = 3x = 3 \times 16 = 48 \, \text{cm}\)

### Conclusion
The length of the longer parallel side is:
\[ \boxed{48 \, \text{cm}} \]