The area of a rhombus is 216 sq. cm. If its one diagonal is 24 cm, find :
length of its side,

Answer

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

### Step 1: Write down 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 into the area formula.
We know the area \( A = 216 \, \text{cm}^2 \) and one diagonal \( d_1 = 24 \, \text{cm} \). We need to find the second diagonal \( d_2 \).
\[ 216 = \frac{1}{2} \times 24 \times d_2 \]

### Step 3: Solve for \( d_2 \).
First, multiply both sides by 2 to eliminate the fraction:
\[ 432 = 24 \times d_2 \]
Now, divide both sides by 24:
\[ d_2 = \frac{432}{24} = 18 \, \text{cm} \]

### Step 4: Use the diagonals to find the length of the side of the rhombus.
The diagonals of a rhombus bisect each other at right angles. Let’s denote the intersection point of the diagonals as \( O \). Therefore, we have:
\[ AO = \frac{d_1}{2} = \frac{24}{2} = 12 \, \text{cm} \]
\[ OC = \frac{d_2}{2} = \frac{18}{2} = 9 \, \text{cm} \]

### Step 5: Apply the Pythagorean theorem to find the length of the side \( AB \).
In triangle \( AOB \), we can use the Pythagorean theorem:
\[ AB^2 = AO^2 + OB^2 \]
Since \( OB \) is equal to \( OC \):
\[ AB^2 = AO^2 + OC^2 \]
Substituting the values:
\[ AB^2 = 12^2 + 9^2 \]
\[ AB^2 = 144 + 81 \]
\[ AB^2 = 225 \]
Taking the square root:
\[ AB = \sqrt{225} = 15 \, \text{cm} \]

### Final Answer:
The length of the side of the rhombus is \( 15 \, \text{cm} \).

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