A diagonal of a quadrilateral is 26 cm and the perpendiculars drawn to it from the opposite vertices are 12.8 cm and 11.2 cm. Find the area of the quadrilateral.

Answer

To find the area of the quadrilateral, we can use the formula for the area of a quadrilateral when the lengths of the diagonals and the perpendiculars from the opposite vertices to the diagonal are known. The area \( A \) can be calculated using the formula:

\[ A = \frac{1}{2} \times d \times (h_1 + h_2) \]

where:
- \( d \) is the length of the diagonal,
- \( h_1 \) and \( h_2 \) are the lengths of the perpendiculars from the opposite vertices to the diagonal.

### Step-by-Step Solution:

1. **Identify the given values**:
- Length of the diagonal \( d = 26 \) cm
- Length of the first perpendicular \( h_1 = 12.8 \) cm
- Length of the second perpendicular \( h_2 = 11.2 \) cm

2. **Substitute the values into the area formula**:
\[ A = \frac{1}{2} \times 26 \times (12.8 + 11.2) \]

3. **Calculate the sum of the perpendiculars**:
\[ h_1 + h_2 = 12.8 + 11.2 = 24 \text{ cm} \]

4. **Now substitute this sum back into the area formula**:
\[ A = \frac{1}{2} \times 26 \times 24 \]

5. **Calculate \( \frac{1}{2} \times 26 \)**:
\[ \frac{1}{2} \times 26 = 13 \]

6. **Now multiply by 24**:
\[ A = 13 \times 24 = 312 \text{ cm}^2 \]

### Final Answer:
The area of the quadrilateral is \( 312 \text{ cm}^2 \).