The length of each side of a rhombus is equal to the length of the side of a square whose diagonal is `40sqrt(2)` cm. If the length of the diagonals of the rhombus are in the ratio 3:4. then its area (in `cm^(2)`) is

Options

A. 1550
B. 1600
C. 1535
D. 1536

Answer

To solve the problem step by step, we need to find the area of a rhombus whose side length is equal to the side of a square with a given diagonal, and whose diagonals are in a specific ratio. Here’s how to approach the problem:

### Step 1: Find the side length of the square
The diagonal of the square is given as \( 40\sqrt{2} \) cm. We know that the relationship between the diagonal \( d \) and the side length \( s \) of a square is given by the formula:
\[ d = s\sqrt{2} \]
Substituting the value of the diagonal:
\[ 40\sqrt{2} = s\sqrt{2} \]
To find \( s \), we can divide both sides by \( \sqrt{2} \):
\[ s = 40 \text{ cm} \]

### Step 2: Determine the side length of the rhombus
Since the length of each side of the rhombus is equal to the length of the side of the square, the side length of the rhombus is also:
\[ \text{Side of rhombus} = 40 \text{ cm} \]

### Step 3: Set up the diagonals of the rhombus
Let the lengths of the diagonals of the rhombus be \( d_1 \) and \( d_2 \). According to the problem, the diagonals are in the ratio \( 3:4 \). We can express the diagonals as:
\[ d_1 = 3x \quad \text{and} \quad d_2 = 4x \]

### Step 4: Use the relationship between the sides and diagonals of the rhombus
The relationship between the sides of a rhombus and its diagonals is given by the formula:
\[ \text{Side}^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Substituting the values of the diagonals:
\[ 40^2 = \left(\frac{3x}{2}\right)^2 + \left(\frac{4x}{2}\right)^2 \]
This simplifies to:
\[ 1600 = \left(\frac{3x}{2}\right)^2 + \left(2x\right)^2 \]
Calculating the squares:
\[ 1600 = \frac{9x^2}{4} + 4x^2 \]
To combine the terms, convert \( 4x^2 \) to have a common denominator:
\[ 4x^2 = \frac{16x^2}{4} \]
Thus:
\[ 1600 = \frac{9x^2 + 16x^2}{4} \]
\[ 1600 = \frac{25x^2}{4} \]

### Step 5: Solve for \( x^2 \)
Multiplying both sides by 4:
\[ 6400 = 25x^2 \]
Dividing by 25:
\[ x^2 = \frac{6400}{25} = 256 \]
Taking the square root:
\[ x = 16 \]

### Step 6: Find the lengths of the diagonals
Now substitute \( x \) back to find \( d_1 \) and \( d_2 \):
\[ d_1 = 3x = 3 \times 16 = 48 \text{ cm} \]
\[ d_2 = 4x = 4 \times 16 = 64 \text{ cm} \]

### Step 7: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Substituting the values of the diagonals:
\[ A = \frac{1}{2} \times 48 \times 64 \]
Calculating:
\[ A = \frac{1}{2} \times 3072 = 1536 \text{ cm}^2 \]

### Final Answer
The area of the rhombus is \( 1536 \text{ cm}^2 \).

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