The area and the length of one of the diagonals of a rhombus is `"54 cm"^2` and 9 cm respectively. Find the length of its other diagonal (in cm).

Options

A. 6
B. 24
C. 12
D. 18

Answer

To find the length of the other diagonal of the rhombus, 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: Identify the known values
From the problem, we know:
- The area \( A = 54 \, \text{cm}^2 \)
- One diagonal \( d_1 = 9 \, \text{cm} \)

We need to find the length of the other diagonal \( d_2 \).

### Step 3: Substitute the known values into the formula
Substituting the known values into the area formula:
\[ 54 = \frac{1}{2} \times 9 \times d_2 \]

### Step 4: Solve for \( d_2 \)
To isolate \( d_2 \), first multiply both sides by 2:
\[ 108 = 9 \times d_2 \]
Next, divide both sides by 9:
\[ d_2 = \frac{108}{9} \]

### Step 5: Calculate \( d_2 \)
Calculating the right side gives:
\[ d_2 = 12 \, \text{cm} \]

### Final Answer
The length of the other diagonal \( d_2 \) is \( 12 \, \text{cm} \).

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