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. 26.8 cm
B. 28.4 cm
C. 30.2 cm
D. 32.6 cm

Answer

To find the length of the diagonal of the quadrilateral, we can use the formula for the area of a quadrilateral in terms of the lengths of the perpendiculars from the opposite vertices to the diagonal. 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,
- \( h_1 \) and \( h_2 \) are the lengths of the perpendiculars from the opposite vertices to the diagonal.

Given:
- Area \( A = 227.2 \, \text{cm}^2 \)
- \( h_1 = 7.2 \, \text{cm} \)
- \( h_2 = 8.8 \, \text{cm} \)

We can rearrange the formula to solve for \( d \):

\[ d = \frac{2A}{h_1 + h_2} \]

Now, let's calculate \( h_1 + h_2 \):

\[ h_1 + h_2 = 7.2 + 8.8 = 16 \, \text{cm} \]

Next, substitute the values into the formula for \( d \):

\[ d = \frac{2 \times 227.2}{16} \]

Calculating the numerator:

\[ 2 \times 227.2 = 454.4 \]

Now, divide by \( 16 \):

\[ d = \frac{454.4}{16} = 28.4 \, \text{cm} \]

Thus, the length of the diagonal is:

\[ \boxed{28.4 \, \text{cm}} \]