The area of trapezium is `384 cm ^2` . If its parallele sides are in the ratio 3:5 and the perpendicular distance between them is 12 cm, the smaller of the parallel sides is

Options

A. 20 cm
B. 24 cm
C. 30 cm
D. 36 cm

Answer

To find the smaller of the parallel sides of the trapezium, we can follow these steps:

### Step 1: Understand the given information
We know:
- The area of the trapezium is \(384 \, \text{cm}^2\).
- The ratio of the parallel sides is \(3:5\).
- The perpendicular distance (height) between the parallel sides is \(12 \, \text{cm}\).

### Step 2: Define the parallel sides
Let the lengths of the parallel sides be:
- Smaller side \(A = 3x\)
- Larger side \(B = 5x\)

### Step 3: Use the area formula for trapezium
The formula for the area of a trapezium is given by:
\[ \text{Area} = \frac{1}{2} \times (A + B) \times h \]
Substituting the known values into the formula:
\[ 384 = \frac{1}{2} \times (3x + 5x) \times 12 \]

### Step 4: Simplify the equation
First, simplify \(3x + 5x\):
\[ 3x + 5x = 8x \]
Now substitute this back into the area formula:
\[ 384 = \frac{1}{2} \times 8x \times 12 \]

### Step 5: Calculate the right-hand side
Calculate \(\frac{1}{2} \times 8 \times 12\):
\[ \frac{1}{2} \times 8 = 4 \]
\[ 4 \times 12 = 48 \]
So, we have:
\[ 384 = 48x \]

### Step 6: Solve for \(x\)
To find \(x\), divide both sides by \(48\):
\[ x = \frac{384}{48} = 8 \]

### Step 7: Find the lengths of the parallel sides
Now substitute \(x\) back to find the lengths of the parallel sides:
- Smaller side \(A = 3x = 3 \times 8 = 24 \, \text{cm}\)
- Larger side \(B = 5x = 5 \times 8 = 40 \, \text{cm}\)

### Step 8: Conclusion
The smaller of the parallel sides is:
\[ \boxed{24 \, \text{cm}} \]

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