The area of a rhombus is 256 square cm. and one of its diagonals is twice the other in length. The length of its larger diagonal is

Options

A. 32 cm
B. 16 cm
C. 48 cm
D. 24 cm

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the diagonals of the rhombus.
Let the lengths of the diagonals be \( D_1 \) and \( D_2 \). According to the problem, one diagonal is twice the length of the other. We can express this relationship as:
\[ D_1 = 2D_2 \]

### Step 2: Use the formula for the area of a rhombus.
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times D_1 \times D_2 \]
We know the area is given as 256 square cm. Therefore, we can write:
\[ 256 = \frac{1}{2} \times D_1 \times D_2 \]

### Step 3: Substitute the relationship of the diagonals into the area formula.
Now, substituting \( D_1 = 2D_2 \) into the area formula:
\[ 256 = \frac{1}{2} \times (2D_2) \times D_2 \]
This simplifies to:
\[ 256 = \frac{1}{2} \times 2D_2^2 \]
\[ 256 = D_2^2 \]

### Step 4: Solve for \( D_2 \).
To find \( D_2 \), we take the square root of both sides:
\[ D_2 = \sqrt{256} \]
\[ D_2 = 16 \, \text{cm} \]

### Step 5: Find \( D_1 \).
Now that we have \( D_2 \), we can find \( D_1 \):
\[ D_1 = 2D_2 = 2 \times 16 = 32 \, \text{cm} \]

### Conclusion:
The length of the larger diagonal \( D_1 \) is \( 32 \, \text{cm} \).

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