The parallel sides of a trapezium are in the ratio 3:4. If the distance between the parallel sides is 9 dm and its area is 126 `dm^(2)`, find the lengths of its parallel sides.

Answer

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

### Step 1: Understand the given information
We know:
- The ratio of the parallel sides is 3:4.
- The distance (height) between the parallel sides is 9 dm.
- The area of the trapezium is 126 dm².

### Step 2: Set up the variables
Let the lengths of the parallel sides be:
- First parallel side = 3x
- Second parallel side = 4x

### Step 3: Use the area formula for a trapezium
The formula for the area \( A \) of a trapezium is given by:
\[ A = \frac{1}{2} \times (h) \times (b_1 + b_2) \]
where \( h \) is the height, \( b_1 \) and \( b_2 \) are the lengths of the parallel sides.

Substituting the known values:
\[ 126 = \frac{1}{2} \times 9 \times (3x + 4x) \]

### Step 4: Simplify the equation
First, simplify the expression inside the parentheses:
\[ 3x + 4x = 7x \]
Now substitute this back into the equation:
\[ 126 = \frac{1}{2} \times 9 \times 7x \]

### Step 5: Solve for \( x \)
Multiply both sides by 2 to eliminate the fraction:
\[ 252 = 9 \times 7x \]
Now simplify:
\[ 252 = 63x \]
Now, divide both sides by 63:
\[ x = \frac{252}{63} = 4 \]

### Step 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 = \( 3x = 3 \times 4 = 12 \) dm
- Second parallel side = \( 4x = 4 \times 4 = 16 \) dm

### Conclusion
The lengths of the parallel sides of the trapezium are:
- First parallel side = 12 dm
- Second parallel side = 16 dm

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