A diagonal of a quadrilateral is 30 cm in length and the lengths of perpendiculars to it from the opposite vertices are 6.8 cm and 9.6 cm. Find the area of the quadrilateral.

Answer

To find the area of the quadrilateral given the length of the diagonal and the lengths of the perpendiculars from the opposite vertices, we can use the formula for the area of a quadrilateral when a diagonal is known:

**Area of Quadrilateral = (1/2) × d × (h1 + h2)**

Where:
- \( d \) = length of the diagonal
- \( h1 \) = length of the perpendicular from one vertex to the diagonal
- \( h2 \) = length of the perpendicular from the opposite vertex to the diagonal

### Step-by-Step Solution:

1. **Identify the given values:**
- Length of the diagonal \( d = 30 \, \text{cm} \)
- Length of the perpendicular from one vertex \( h1 = 6.8 \, \text{cm} \)
- Length of the perpendicular from the opposite vertex \( h2 = 9.6 \, \text{cm} \)

**Hint:** Make sure you write down all the values given in the problem.

2. **Substitute the values into the area formula:**
\[ \text{Area} = \frac{1}{2} \times d \times (h1 + h2) \]
\[ \text{Area} = \frac{1}{2} \times 30 \times (6.8 + 9.6) \]

**Hint:** Remember to add \( h1 \) and \( h2 \) together before multiplying.

3. **Calculate \( h1 + h2 \):**
\[ h1 + h2 = 6.8 + 9.6 = 16.4 \, \text{cm} \]

**Hint:** Double-check your addition to ensure accuracy.

4. **Now substitute back into the area formula:**
\[ \text{Area} = \frac{1}{2} \times 30 \times 16.4 \]

**Hint:** You can simplify the calculation by first calculating \( \frac{1}{2} \times 30 \).

5. **Calculate \( \frac{1}{2} \times 30 \):**
\[ \frac{1}{2} \times 30 = 15 \]

**Hint:** This step simplifies the multiplication.

6. **Now calculate the area:**
\[ \text{Area} = 15 \times 16.4 \]
\[ \text{Area} = 246 \, \text{cm}^2 \]

**Hint:** Make sure to multiply carefully to avoid mistakes.

### Final Answer:
The area of the quadrilateral is \( 246 \, \text{cm}^2 \).