If the perimeter of a rhombus is 4p and length of its diagonals are a and b , then its area is :

Options

A. `a/b`
B. `(ab)/2`
C. `(ab)/p`
D. `p(a^(2) + b^(2) )`

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 is a type of polygon that is a quadrilateral (four sides) with all sides having equal length. The diagonals of a rhombus bisect each other at right angles.

### Step 2: Use the perimeter to find the side length
The perimeter of a rhombus is given by the formula:
\[ \text{Perimeter} = 4 \times \text{side length} \]
Given that the perimeter is \(4p\), we can find the side length:
\[ 4 \times \text{side length} = 4p \implies \text{side length} = p \]

### Step 3: Use the formula for the area of a rhombus
The area \(A\) of a rhombus can be calculated using the lengths of its diagonals \(a\) and \(b\) with the formula:
\[ A = \frac{1}{2} \times a \times b \]

### Step 4: Substitute the values into the area formula
Now that we have the lengths of the diagonals \(a\) and \(b\), we can substitute these values into the area formula:
\[ A = \frac{1}{2} \times a \times b \]

### Conclusion
Thus, the area of the rhombus is:
\[ \text{Area} = \frac{1}{2}ab \]

### Answer
The correct option for the area of the rhombus is \( \frac{1}{2}ab \).

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