The area of a trapezium is `72 cm^(2)` and its height is 12 cm . If one of the parallel sides is longer than the other by 2 cm , then find the length of the two parallel sides .

Answer

To solve the problem, we will use the formula for the area of a trapezium. The area \( A \) of a trapezium can be calculated using the formula:

\[ A = \frac{1}{2} \times (a + b) \times h \]

where \( a \) and \( b \) are the lengths of the two parallel sides, and \( h \) is the height of the trapezium.

### Step 1: Identify the given values
- Area \( A = 72 \, \text{cm}^2 \)
- Height \( h = 12 \, \text{cm} \)
- Let the length of the shorter parallel side be \( x \, \text{cm} \).
- The longer parallel side will then be \( x + 2 \, \text{cm} \).

### Step 2: Substitute the values into the area formula
Using the area formula, we can substitute the known values:

\[ 72 = \frac{1}{2} \times (x + (x + 2)) \times 12 \]

### Step 3: Simplify the equation
First, simplify the expression inside the parentheses:

\[ 72 = \frac{1}{2} \times (2x + 2) \times 12 \]

Now, simplify further:

\[ 72 = (x + 1) \times 12 \]

### Step 4: Divide both sides by 12
To isolate \( x + 1 \), divide both sides by 12:

\[ \frac{72}{12} = x + 1 \]

This gives:

\[ 6 = x + 1 \]

### Step 5: Solve for \( x \)
Now, subtract 1 from both sides to find \( x \):

\[ x = 6 - 1 \]

Thus,

\[ x = 5 \, \text{cm} \]

### Step 6: Find the lengths of the two parallel sides
Now that we have \( x \), we can find the lengths of the two parallel sides:

- Shorter side: \( x = 5 \, \text{cm} \)
- Longer side: \( x + 2 = 5 + 2 = 7 \, \text{cm} \)

### Final Answer
The lengths of the two parallel sides are:
- Shorter side: \( 5 \, \text{cm} \)
- Longer side: \( 7 \, \text{cm} \)

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