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: Identify the given values
- Perimeter of the trapezium (P) = 104 cm
- Length of non-parallel sides (L1 and L2) = 18 cm and 22 cm
- Altitude (height) (h) = 16 cm

### Step 2: Calculate the sum of the non-parallel sides
The sum of the non-parallel sides is:
\[ L1 + L2 = 18 \, \text{cm} + 22 \, \text{cm} = 40 \, \text{cm} \]

### Step 3: Use the perimeter to find the sum of the parallel sides
The formula for the perimeter of a trapezium is:
\[ P = L1 + L2 + \text{(sum of parallel sides)} \]
Let the sum of the parallel sides be \( S \).
So, we can write:
\[ 104 = 40 + S \]
To find \( S \):
\[ S = 104 - 40 = 64 \, \text{cm} \]

### Step 4: Use the area formula for the trapezium
The area \( A \) of a trapezium can be calculated using the formula:
\[ A = \frac{1}{2} \times (b1 + b2) \times h \]
Where \( b1 \) and \( b2 \) are the lengths of the parallel sides and \( h \) is the height.

Since we have the sum of the parallel sides \( S = b1 + b2 = 64 \, \text{cm} \) and the height \( h = 16 \, \text{cm} \), we can substitute these values into the area formula:
\[ A = \frac{1}{2} \times 64 \times 16 \]

### Step 5: Calculate the area
Now, we can compute the area:
\[ A = \frac{1}{2} \times 64 \times 16 = 32 \times 16 = 512 \, \text{cm}^2 \]

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

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