Area of rhombus is `256 cm^2`. One of the diagonal is half of the other diagonal. The sum of the diagonals is :

Options

A. 38 cm
B. 48 cm
C. 28 cm
D. 56 cm

Answer

To solve the problem step by step, we will use the properties of a rhombus and the given information.

### Step 1: Understand the relationship between the diagonals
Let the length of one diagonal be \( d_1 \) and the other diagonal be \( d_2 \). According to the problem, one diagonal is half of the other. We can express this relationship as:
\[ d_2 = \frac{1}{2} d_1 \]

### Step 2: Use the area formula for a rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Given that the area is \( 256 \, \text{cm}^2 \), we can substitute the values:
\[ 256 = \frac{1}{2} \times d_1 \times d_2 \]

### Step 3: Substitute the relationship of the diagonals into the area formula
Substituting \( d_2 = \frac{1}{2} d_1 \) into the area formula:
\[ 256 = \frac{1}{2} \times d_1 \times \left(\frac{1}{2} d_1\right) \]
This simplifies to:
\[ 256 = \frac{1}{4} d_1^2 \]

### Step 4: Solve for \( d_1 \)
To isolate \( d_1^2 \), multiply both sides by 4:
\[ 1024 = d_1^2 \]
Now, take the square root of both sides:
\[ d_1 = \sqrt{1024} = 32 \, \text{cm} \]

### Step 5: Find \( d_2 \)
Now that we have \( d_1 \), we can find \( d_2 \):
\[ d_2 = \frac{1}{2} d_1 = \frac{1}{2} \times 32 = 16 \, \text{cm} \]

### Step 6: Calculate the sum of the diagonals
Now we can find the sum of the diagonals:
\[ \text{Sum of the diagonals} = d_1 + d_2 = 32 + 16 = 48 \, \text{cm} \]

### Final Answer
The sum of the diagonals is \( 48 \, \text{cm} \).

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