The area of a trapezium is `216 m^(2)` and its height is 12m. If one of the parallel sides is 14 m less than the other, find the length of each of the parallel sides.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We know the area of the trapezium (A) is 216 m², the height (h) is 12 m, and one parallel side is 14 m less than the other.

### Step 2: Set Up the Variables
Let the length of the longer parallel side be \( x \) meters. Then, the length of the shorter parallel side will be \( x - 14 \) meters.

### Step 3: Use the Area Formula
The formula for the area of a trapezium is given by:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \( b_1 \) and \( b_2 \) are the lengths of the parallel sides. Substituting the values we have:
\[ 216 = \frac{1}{2} \times (x + (x - 14)) \times 12 \]

### Step 4: Simplify the Equation
First, simplify the expression inside the parentheses:
\[ x + (x - 14) = 2x - 14 \]
Now substituting this back into the area formula:
\[ 216 = \frac{1}{2} \times (2x - 14) \times 12 \]
Now, simplify further:
\[ 216 = (2x - 14) \times 6 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 6:
\[ 36 = 2x - 14 \]
Next, add 14 to both sides:
\[ 36 + 14 = 2x \]
\[ 50 = 2x \]
Now, divide both sides by 2:
\[ x = 25 \]

### Step 6: Find the Length of Each Parallel Side
Now that we have \( x \), we can find the lengths of the parallel sides:
- The longer side \( AD = x = 25 \) m
- The shorter side \( BC = x - 14 = 25 - 14 = 11 \) m

### Final Answer
The lengths of the parallel sides are:
- Longer side \( AD = 25 \) m
- Shorter side \( BC = 11 \) m

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