The diagonals of a rhombus are in the ratio 3 : 4. If the perimeter of the rhombus is 40 cm, then find its area.

Options

A. `96 cm^2`
B. `192 cm^2`
C. `84 cm^2`
D. `112 cm^2`

Answer

To solve the problem step by step, we will follow the given information about the rhombus and apply the necessary formulas.

### Step 1: Understand the given information
- The diagonals of the rhombus are in the ratio 3:4.
- The perimeter of the rhombus is 40 cm.

### Step 2: Find the length of one side of the rhombus
The perimeter \( P \) of a rhombus is given by the formula:
\[ P = 4a \]
where \( a \) is the length of one side. Given that the perimeter is 40 cm:
\[ 4a = 40 \]
Now, solving for \( a \):
\[ a = \frac{40}{4} = 10 \text{ cm} \]

### Step 3: Express the diagonals in terms of a variable
Let the lengths of the diagonals be \( D_1 \) and \( D_2 \). Given the ratio of the diagonals:
\[ \frac{D_1}{D_2} = \frac{3}{4} \]
We can express \( D_1 \) in terms of \( D_2 \):
\[ D_1 = \frac{3}{4}D_2 \]

### Step 4: Use the property of the rhombus
The diagonals of a rhombus bisect each other at right angles. Therefore, we can use the Pythagorean theorem in one of the triangles formed by the diagonals. The half lengths of the diagonals are:
\[ \frac{D_1}{2} \text{ and } \frac{D_2}{2} \]
Using the Pythagorean theorem:
\[ a^2 = \left(\frac{D_1}{2}\right)^2 + \left(\frac{D_2}{2}\right)^2 \]
Substituting \( a = 10 \):
\[ 10^2 = \left(\frac{D_1}{2}\right)^2 + \left(\frac{D_2}{2}\right)^2 \]
This simplifies to:
\[ 100 = \frac{D_1^2}{4} + \frac{D_2^2}{4} \]
Multiplying through by 4 to eliminate the fractions:
\[ 400 = D_1^2 + D_2^2 \]

### Step 5: Substitute \( D_1 \) into the equation
Now substitute \( D_1 = \frac{3}{4}D_2 \) into the equation:
\[ 400 = \left(\frac{3}{4}D_2\right)^2 + D_2^2 \]
This expands to:
\[ 400 = \frac{9}{16}D_2^2 + D_2^2 \]
Combining the terms:
\[ 400 = \frac{9}{16}D_2^2 + \frac{16}{16}D_2^2 = \frac{25}{16}D_2^2 \]
Now, multiply both sides by 16:
\[ 6400 = 25D_2^2 \]
Dividing by 25:
\[ D_2^2 = \frac{6400}{25} = 256 \]
Taking the square root:
\[ D_2 = 16 \text{ cm} \]

### Step 6: Find \( D_1 \)
Now substitute back to find \( D_1 \):
\[ D_1 = \frac{3}{4}D_2 = \frac{3}{4} \times 16 = 12 \text{ cm} \]

### Step 7: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{D_1 \times D_2}{2} \]
Substituting the values of \( D_1 \) and \( D_2 \):
\[ A = \frac{12 \times 16}{2} = \frac{192}{2} = 96 \text{ cm}^2 \]

### Final Answer
The area of the rhombus is \( 96 \text{ cm}^2 \).