The area of a rhombus is `216 cm^(2)` and the length of one of its diagonals is 24 cm. How long is each side of the rhombus ?

Options

A. 14 cm
B. 13 cm
C. 15 cm
D. 12 cm

Answer

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

### Step 1: 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 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 can plug these values into the formula:
\[ 216 = \frac{1}{2} \times 24 \times d_2 \]

### Step 3: Solve for the second diagonal \( d_2 \).
First, simplify the equation:
\[ 216 = 12 \times d_2 \]
Now, divide both sides by 12:
\[ d_2 = \frac{216}{12} = 18 \, \text{cm} \]

### Step 4: Find the lengths of the halves of the diagonals.
Since the diagonals bisect each other at right angles, we can find the lengths of the halves:
- Half of \( d_1 \) (which is \( 24 \, \text{cm} \)):
\[ \frac{d_1}{2} = \frac{24}{2} = 12 \, \text{cm} \]
- Half of \( d_2 \) (which is \( 18 \, \text{cm} \)):
\[ \frac{d_2}{2} = \frac{18}{2} = 9 \, \text{cm} \]

### Step 5: Use the Pythagorean theorem to find the side of the rhombus.
In the right triangle formed by the halves of the diagonals, we can use the Pythagorean theorem:
\[ s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Substituting the values we found:
\[ s^2 = 12^2 + 9^2 \]
Calculating the squares:
\[ s^2 = 144 + 81 = 225 \]

### Step 6: Solve for \( s \).
Taking the square root of both sides:
\[ s = \sqrt{225} = 15 \, \text{cm} \]

### Conclusion
The length of each side of the rhombus is \( 15 \, \text{cm} \).

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