Sides of a triangle are in the ratio `12:17:25` and its perimeter is `540 cm.` Its area will be-

Options

A. `6000cm^(2)`
B. `9000cm^(2)`
C. `12000cm^(2)`
D. None of these

Answer

To find the area of a triangle with sides in the ratio 12:17:25 and a perimeter of 540 cm, we can follow these steps:

### Step 1: Determine the sides of the triangle
Let the sides of the triangle be represented as:
- \( A = 12k \)
- \( B = 17k \)
- \( C = 25k \)

Given that the perimeter is 540 cm, we can set up the equation:
\[ A + B + C = 540 \]
Substituting the values:
\[ 12k + 17k + 25k = 540 \]

### Step 2: Solve for \( k \)
Combine the terms:
\[ 54k = 540 \]
Now, divide both sides by 54:
\[ k = \frac{540}{54} = 10 \]

### Step 3: Calculate the lengths of the sides
Now, substitute \( k \) back into the expressions for the sides:
- \( A = 12k = 12 \times 10 = 120 \, \text{cm} \)
- \( B = 17k = 17 \times 10 = 170 \, \text{cm} \)
- \( C = 25k = 25 \times 10 = 250 \, \text{cm} \)

### Step 4: Calculate the semi-perimeter
The semi-perimeter \( s \) is given by:
\[ s = \frac{A + B + C}{2} = \frac{540}{2} = 270 \, \text{cm} \]

### Step 5: Apply Heron's formula to find the area
Heron's formula for the area \( A \) of the triangle is:
\[ \text{Area} = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values we have:
\[ \text{Area} = \sqrt{270 \times (270 - 120) \times (270 - 170) \times (270 - 250)} \]
Calculating each term:
- \( s - A = 270 - 120 = 150 \)
- \( s - B = 270 - 170 = 100 \)
- \( s - C = 270 - 250 = 20 \)

Now substituting these values back into the area formula:
\[ \text{Area} = \sqrt{270 \times 150 \times 100 \times 20} \]

### Step 6: Simplify the expression under the square root
Calculating the product:
\[ 270 \times 150 = 40500 \]
\[ 40500 \times 100 = 4050000 \]
\[ 4050000 \times 20 = 81000000 \]

So we have:
\[ \text{Area} = \sqrt{81000000} \]
Breaking it down:
\[ \sqrt{81000000} = \sqrt{81 \times 10^6} = 9 \times 1000 = 9000 \, \text{cm}^2 \]

### Final Answer
Thus, the area of the triangle is:
\[ \text{Area} = 9000 \, \text{cm}^2 \]