The lengths of parallel sides of a trapezium are in the ratio `3 : 5` and the distance between them is `10 cm`. If the area of the trapezium is `120 cm^(2)`, find the lengths of its parallel sides.

Options

A. `9cm` and `17cm`
B. `8cm` and `15 cm`
C. `9cm` and `15 cm`
D. `5cm` and `17 cm`

Answer

To find the lengths of the parallel sides of the trapezium, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Ratio of the Parallel Sides**:
The lengths of the parallel sides are given in the ratio of 3:5. We can represent these sides as:
- Let the first parallel side be \(3x\).
- Let the second parallel side be \(5x\).

2. **Write the Area Formula for a Trapezium**:
The formula for the area \(A\) of a trapezium is:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \(b_1\) and \(b_2\) are the lengths of the parallel sides and \(h\) is the height (distance between the parallel sides).

3. **Substitute the Known Values into the Formula**:
We know the area \(A = 120 \, \text{cm}^2\) and the distance between the parallel sides \(h = 10 \, \text{cm}\). Substituting these values into the area formula gives:
\[ 120 = \frac{1}{2} \times (3x + 5x) \times 10 \]

4. **Simplify the Equation**:
Combine the terms inside the parentheses:
\[ 120 = \frac{1}{2} \times (8x) \times 10 \]
This simplifies to:
\[ 120 = 40x \]

5. **Solve for \(x\)**:
To find \(x\), divide both sides by 40:
\[ x = \frac{120}{40} = 3 \]

6. **Find the Lengths of the Parallel Sides**:
Now that we have \(x\), we can find the lengths of the parallel sides:
- First parallel side:
\[ 3x = 3 \times 3 = 9 \, \text{cm} \]
- Second parallel side:
\[ 5x = 5 \times 3 = 15 \, \text{cm} \]

### Final Answer:
The lengths of the parallel sides are:
- First parallel side: \(9 \, \text{cm}\)
- Second parallel side: \(15 \, \text{cm}\)

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