A quadrilateral has a diagonal of 26 cm. The lengths of perpendiculars on this diagonal from opposite vertices is 13 cm and 7 cm. Find the area of this quadrilateral.

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length of the diagonal (d) = 26 cm
- Length of the perpendicular from one vertex (h1) = 13 cm
- Length of the perpendicular from the opposite vertex (h2) = 7 cm

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

3. **Substitute the Values into the Formula:**
Now, substitute the values we identified:
\[ A = \frac{1}{2} \times 26 \times (13 + 7) \]

4. **Calculate the Sum of the Perpendiculars:**
First, calculate \(h1 + h2\):
\[ h1 + h2 = 13 + 7 = 20 \]

5. **Calculate the Area:**
Now substitute back into the area formula:
\[ A = \frac{1}{2} \times 26 \times 20 \]
- Calculate \(\frac{1}{2} \times 26 = 13\)
- Then, calculate \(13 \times 20 = 260\)

6. **Final Result:**
Therefore, the area of the quadrilateral is:
\[ A = 260 \, \text{cm}^2 \]