The area of a rhombus is `216 cm^(2)` and the length of its one diagonal is 24 cm. The perimeter (in cm) of the rhombus is

Options

A. 52
B. 60
C. 120
D. 100

Answer

To find the perimeter of the rhombus given its area and one diagonal, we can follow these steps:

### Step 1: Understand 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 2: Substitute the known values
We know the area \( A = 216 \, \text{cm}^2 \) and one diagonal \( d_1 = 24 \, \text{cm} \). We need to find the other diagonal \( d_2 \).
\[ 216 = \frac{1}{2} \times 24 \times d_2 \]

### Step 3: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ 432 = 24 \times d_2 \]

### Step 4: Solve for \( d_2 \)
Now, divide both sides by 24:
\[ d_2 = \frac{432}{24} = 18 \, \text{cm} \]

### Step 5: Calculate the lengths of the sides of the rhombus
The diagonals of a rhombus bisect each other at right angles. Therefore, we can find the length of each side using the Pythagorean theorem. Each half of the diagonals will be:
\[ \frac{d_1}{2} = \frac{24}{2} = 12 \, \text{cm} \]
\[ \frac{d_2}{2} = \frac{18}{2} = 9 \, \text{cm} \]
Now, we can calculate the length of one side \( s \) of the rhombus:
\[ s = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} = \sqrt{12^2 + 9^2} \]
\[ s = \sqrt{144 + 81} = \sqrt{225} = 15 \, \text{cm} \]

### Step 6: Calculate the perimeter of the rhombus
The perimeter \( P \) of a rhombus is given by:
\[ P = 4 \times s \]
Substituting the value of \( s \):
\[ P = 4 \times 15 = 60 \, \text{cm} \]

### Final Answer
The perimeter of the rhombus is \( 60 \, \text{cm} \).

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