The length of the diagonals of a rhombus is in the ratio 4:3. If its area is 384 `cm^(2)`, find its side.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the problem
The problem states that the diagonals of a rhombus are in the ratio of 4:3 and that the area of the rhombus is 384 cm². We need to find the length of one side of the rhombus.

### Step 2: Assign variables to the diagonals
Let the lengths of the diagonals be:
- Larger diagonal \( d_1 = 4x \)
- Smaller diagonal \( d_2 = 3x \)

### Step 3: Use the area formula for a rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Substituting the values of the diagonals:
\[ 384 = \frac{1}{2} \times (4x) \times (3x) \]

### Step 4: Simplify the equation
\[ 384 = \frac{1}{2} \times 12x^2 \]
\[ 384 = 6x^2 \]

### Step 5: Solve for \( x^2 \)
To find \( x^2 \), divide both sides by 6:
\[ x^2 = \frac{384}{6} = 64 \]

### Step 6: Find \( x \)
Taking the square root of both sides:
\[ x = \sqrt{64} = 8 \text{ cm} \]

### Step 7: Calculate the lengths of the diagonals
Now, we can find the lengths of the diagonals:
- \( d_1 = 4x = 4 \times 8 = 32 \text{ cm} \)
- \( d_2 = 3x = 3 \times 8 = 24 \text{ cm} \)

### Step 8: Use Pythagorean theorem to find the side of the rhombus
The diagonals of a rhombus bisect each other at right angles. Therefore, we can use the Pythagorean theorem to find the length of one side \( s \) of the rhombus.

The half lengths of the diagonals are:
- Half of \( d_1 = \frac{32}{2} = 16 \text{ cm} \)
- Half of \( d_2 = \frac{24}{2} = 12 \text{ cm} \)

Using the Pythagorean theorem:
\[ s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
\[ s^2 = 16^2 + 12^2 \]
\[ s^2 = 256 + 144 = 400 \]
\[ s = \sqrt{400} = 20 \text{ cm} \]

### Final Answer
The length of each side of the rhombus is \( 20 \text{ cm} \).

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