The lengths of the sides of a triangle are in the ratio `4 : 5 : 3` and its perimeter is `96 cm.` Find its area.

Options

A. `"300 cm"^(2)`
B. `"380 cm"^(2)`
C. `"384 cm"^(2)`
D. `"364 cm"^(2)`

Explanation

To find the area of the triangle with sides in the ratio 4:5:3 and a perimeter of 96 cm, we can follow these steps:

### Step 1: Determine the lengths of the sides
Given the ratio of the sides is 4:5:3, we can express the sides in terms of a variable \( x \):
- Let the sides be \( 4x, 5x, \) and \( 3x \).

### Step 2: Set up the equation for the perimeter
The perimeter of the triangle is the sum of its sides:
\[ 4x + 5x + 3x = 96 \text{ cm} \]
This simplifies to:
\[ 12x = 96 \]

### Step 3: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{96}{12} = 8 \]

### Step 4: Calculate the lengths of the sides
Now we can find the actual lengths of the sides:
- Side A: \( 4x = 4 \times 8 = 32 \text{ cm} \)
- Side B: \( 5x = 5 \times 8 = 40 \text{ cm} \)
- Side C: \( 3x = 3 \times 8 = 24 \text{ cm} \)

### Step 5: Calculate the semi-perimeter
The semi-perimeter \( s \) is half of the perimeter:
\[ s = \frac{96}{2} = 48 \text{ cm} \]

### Step 6: Apply Heron's formula
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( a, b, c \) are the lengths of the sides. Plugging in the values:
- \( a = 32 \text{ cm} \)
- \( b = 40 \text{ cm} \)
- \( c = 24 \text{ cm} \)

We calculate:
\[ A = \sqrt{48(48-32)(48-40)(48-24)} \]
Calculating each term:
- \( s - a = 48 - 32 = 16 \)
- \( s - b = 48 - 40 = 8 \)
- \( s - c = 48 - 24 = 24 \)

Now substituting these values:
\[ A = \sqrt{48 \times 16 \times 8 \times 24} \]

### Step 7: Simplify the expression
Calculating the product:
\[ 48 \times 16 = 768 \]
\[ 768 \times 8 = 6144 \]
\[ 6144 \times 24 = 147456 \]

Now, taking the square root:
\[ A = \sqrt{147456} = 384 \text{ cm}^2 \]

### Final Answer
The area of the triangle is \( 384 \text{ cm}^2 \).

---