The perimeter of a trapezium is 104 cm, the lengths of its non-parallel sides are 18 cm and 22 cm and its altitude is 16 cm . The area (in ` cm^(2)` ) of the trapezium is

Options

A. 1024
B. 640
C. 512
D. 320

Answer

To find the area of the trapezium, we can follow these steps:

### Step 1: Understand the given information
We know the following:
- Perimeter of the trapezium = 104 cm
- Length of non-parallel sides (let's call them L1 and L2) = 18 cm and 22 cm
- Altitude (height) of the trapezium = 16 cm

### Step 2: Set up the equation for the perimeter
The perimeter of a trapezium is given by the formula:
\[ \text{Perimeter} = L1 + L2 + a + b \]
where \( a \) and \( b \) are the lengths of the parallel sides.

Substituting the known values:
\[ 104 = 18 + 22 + a + b \]

### Step 3: Simplify the equation
Combine the lengths of the non-parallel sides:
\[ 104 = 40 + a + b \]

### Step 4: Solve for the sum of the parallel sides
Rearranging the equation gives:
\[ a + b = 104 - 40 \]
\[ a + b = 64 \]

### Step 5: Use the area formula for the trapezium
The area \( A \) of a trapezium can be calculated using the formula:
\[ A = \frac{1}{2} \times (a + b) \times \text{height} \]

Substituting the values we have:
\[ A = \frac{1}{2} \times 64 \times 16 \]

### Step 6: Calculate the area
Calculating the area:
\[ A = 32 \times 16 \]
\[ A = 512 \, \text{cm}^2 \]

Thus, the area of the trapezium is \( 512 \, \text{cm}^2 \).

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