If the perimeter of a rhombus is 4a and the lengths of the diagonals are x and y, then its area is

Options

A. `a(x+y)`
B. `x^(2)+y^(2)`
C. xy
D. `1/(2)xy`

Answer

To find the area of a rhombus given its perimeter and the lengths of its diagonals, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the properties of a rhombus**: A rhombus is a type of polygon that is a quadrilateral (four sides) with all sides of equal length. The diagonals of a rhombus bisect each other at right angles.

2. **Identify the given information**: We are given that the perimeter of the rhombus is \(4a\) and the lengths of the diagonals are \(x\) and \(y\).

3. **Relate the perimeter to the side length**: The perimeter \(P\) of a rhombus can be expressed as:
\[ P = 4s \]
where \(s\) is the length of one side. Given that the perimeter is \(4a\), we can equate:
\[ 4s = 4a \]
Dividing both sides by 4 gives:
\[ s = a \]

4. **Use the formula for the area of a rhombus**: The area \(A\) of a rhombus can be calculated using the lengths of its diagonals \(d_1\) and \(d_2\):
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
In our case, \(d_1 = x\) and \(d_2 = y\).

5. **Substitute the values into the area formula**: Plugging in the values of the diagonals, we get:
\[ A = \frac{1}{2} \times x \times y \]

6. **Final answer**: Thus, the area of the rhombus is:
\[ A = \frac{1}{2}xy \]

### Summary:
The area of the rhombus, given the perimeter of \(4a\) and the diagonals \(x\) and \(y\), is:
\[ \text{Area} = \frac{1}{2}xy \]