Find the area of a quadrilateral one of whose diagonals is 40 cm and the lengths of the perpendiculars drawn from the opposite vertices on the diagonal are 16 cm and 12 cm.

Answer

To find the area of the quadrilateral given the length of one of its diagonals and the lengths of the perpendiculars from the opposite vertices to that diagonal, we can use the following steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- Length of the diagonal (D) = 40 cm
- Length of the perpendicular from one vertex (h1) = 16 cm
- Length of the perpendicular from the opposite vertex (h2) = 12 cm

2. **Use the formula for the area of the quadrilateral**:
The area (A) of a quadrilateral when one diagonal and the lengths of the perpendiculars from the opposite vertices are known can be calculated using the formula:
\[ A = \frac{1}{2} \times D \times (h1 + h2) \]

3. **Substitute the values into the formula**:
\[ A = \frac{1}{2} \times 40 \times (16 + 12) \]

4. **Calculate the sum of the perpendiculars**:
\[ h1 + h2 = 16 + 12 = 28 \]

5. **Substitute the sum back into the area formula**:
\[ A = \frac{1}{2} \times 40 \times 28 \]

6. **Calculate the area**:
\[ A = 20 \times 28 = 560 \text{ cm}^2 \]

7. **Final result**:
The area of the quadrilateral is **560 cm²**.