The area of a quadrilateral is `227.2 cm^(2)` and the length of the perpendiculars from the opposite vertices to a diagonal are `7.2 cm` and `8 . 8 cm`. What is the length of the diagonal ?

Options

A. `33.6 cm`
B. `26.8 cm`
C. `28.4 cm`
D. `30. 2 cm`

Answer

To find the length of the diagonal of the quadrilateral given the area and the lengths of the perpendiculars from the opposite vertices to the diagonal, we can use the formula for the area of a quadrilateral in terms of its diagonal and the heights from the opposite vertices.

### Step-by-Step Solution:

1. **Understand the Formula**: The area \( A \) of a quadrilateral can be expressed as:
\[ A = \frac{1}{2} \times d \times (h_1 + h_2) \]
where \( d \) is the length of the diagonal, \( h_1 \) is the length of the perpendicular from one vertex to the diagonal, and \( h_2 \) is the length of the perpendicular from the opposite vertex to the diagonal.

2. **Substitute the Given Values**: We know:
- Area \( A = 227.2 \, \text{cm}^2 \)
- \( h_1 = 7.2 \, \text{cm} \)
- \( h_2 = 8.8 \, \text{cm} \)

Plugging these values into the formula gives:
\[ 227.2 = \frac{1}{2} \times d \times (7.2 + 8.8) \]

3. **Calculate the Sum of Heights**: First, calculate \( h_1 + h_2 \):
\[ h_1 + h_2 = 7.2 + 8.8 = 16 \, \text{cm} \]

4. **Substitute Back into the Area Formula**: Now substitute this sum back into the area formula:
\[ 227.2 = \frac{1}{2} \times d \times 16 \]

5. **Simplify the Equation**: Multiply both sides by 2 to eliminate the fraction:
\[ 454.4 = d \times 16 \]

6. **Solve for \( d \)**: Now, divide both sides by 16 to find \( d \):
\[ d = \frac{454.4}{16} \]

7. **Calculate the Length of the Diagonal**:
\[ d = 28.4 \, \text{cm} \]

### Final Answer:
The length of the diagonal is \( 28.4 \, \text{cm} \).