The area of a trapezium is 384 cm . If its parallel 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:
- Area of the trapezium = 384 cm²
- The ratio of the parallel sides = 3:5
- The height (perpendicular distance between the parallel sides) = 12 cm

### Step 2: Assign variables to the parallel sides
Let the lengths of the parallel sides be:
- Smaller side = 3x
- Larger side = 5x

### Step 3: Use the area formula for 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.

Substituting the known values into the formula:
\[ 384 = \frac{1}{2} \times (3x + 5x) \times 12 \]

### Step 4: Simplify the equation
First, simplify the expression in the parentheses:
\[ 384 = \frac{1}{2} \times (8x) \times 12 \]
Now, calculate \( \frac{1}{2} \times 12 = 6 \):
\[ 384 = 6 \times 8x \]
This simplifies to:
\[ 384 = 48x \]

### Step 5: Solve for \( x \)
To find \( x \), divide both sides by 48:
\[ x = \frac{384}{48} \]
Calculating this gives:
\[ x = 8 \]

### Step 6: Find the lengths of the parallel sides
Now that we have \( x \), we can find the lengths of the parallel sides:
- Smaller side = \( 3x = 3 \times 8 = 24 \) cm
- Larger side = \( 5x = 5 \times 8 = 40 \) cm

### Conclusion
The smaller of the parallel sides is **24 cm**.

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