Find the area of a quadrilateral which has one diagonal measuring 22 cm, and the perpendiculars from the opposite vertices on this diagonal are 6 cm and 10 cm, respectively.

Answer

To find the area of the quadrilateral given the diagonal and the perpendicular heights from the opposite vertices, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Diagonal (D1) = 22 cm
- Height from one vertex (h1) = 6 cm
- Height from the opposite vertex (h2) = 10 cm

2. **Use the Area Formula for a Quadrilateral:**
The area \( A \) of a quadrilateral can be calculated using the formula:
\[ A = \frac{1}{2} \times D1 \times (h1 + h2) \]
where \( D1 \) is the length of the diagonal, and \( h1 \) and \( h2 \) are the heights from the opposite vertices to the diagonal.

3. **Substitute the Values into the Formula:**
Plugging in the values we have:
\[ A = \frac{1}{2} \times 22 \times (6 + 10) \]

4. **Calculate the Sum of Heights:**
First, calculate \( h1 + h2 \):
\[ h1 + h2 = 6 + 10 = 16 \]

5. **Calculate the Area:**
Now substitute back into the area formula:
\[ A = \frac{1}{2} \times 22 \times 16 \]
Simplifying this:
\[ A = 11 \times 16 = 176 \text{ cm}^2 \]

6. **Final Answer:**
Therefore, the area of the quadrilateral is:
\[ \text{Area} = 176 \text{ cm}^2 \]