If the perimeter of a rhombus is `4a` and the length 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 1: Understand the properties of a rhombus
A rhombus has four equal sides, and its diagonals bisect each other at right angles. The area of a rhombus can be calculated using the lengths of its diagonals.

### Step 2: Use the formula for the area of a rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

### Step 3: Substitute the values of the diagonals
In this case, the diagonals are given as \( x \) and \( y \). Therefore, we can substitute these values into the area formula:
\[ A = \frac{1}{2} \times x \times y \]

### Step 4: Finalize the area expression
Thus, the area of the rhombus can be expressed as:
\[ A = \frac{1}{2} xy \]

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

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