The perimeter of a triangle is 450 m and its sides are in the ratio `12 : 5 : 13`. Find the area of the triangle.

Answer

To find the area of the triangle given its perimeter and the ratio of its sides, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the sides in terms of a variable:**
Given the ratio of the sides of the triangle is \(12:5:13\), we can express the sides as:
- Side A = \(12x\)
- Side B = \(5x\)
- Side C = \(13x\)

2. **Set up the equation for the perimeter:**
The perimeter of the triangle is the sum of its sides:
\[ A + B + C = 450 \text{ m} \]
Substituting the expressions for the sides:
\[ 12x + 5x + 13x = 450 \]

3. **Combine like terms:**
Combine the terms on the left side:
\[ 30x = 450 \]

4. **Solve for \(x\):**
Divide both sides by 30:
\[ x = \frac{450}{30} = 15 \text{ m} \]

5. **Calculate the lengths of the sides:**
Now substitute \(x\) back into the expressions for the sides:
- Side A = \(12 \times 15 = 180 \text{ m}\)
- Side B = \(5 \times 15 = 75 \text{ m}\)
- Side C = \(13 \times 15 = 195 \text{ m}\)

6. **Calculate the semi-perimeter (s):**
The semi-perimeter \(s\) is given by:
\[ s = \frac{A + B + C}{2} = \frac{450}{2} = 225 \text{ m} \]

7. **Use Heron's formula to find the area:**
Heron's formula states that the area \(A\) of the triangle can be calculated as:
\[ A = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ A = \sqrt{225(225 - 180)(225 - 75)(225 - 195)} \]
Simplifying the terms inside the square root:
\[ A = \sqrt{225 \times 45 \times 150 \times 30} \]

8. **Calculate the area:**
First, calculate the product:
\[ 225 \times 45 = 10125 \]
\[ 150 \times 30 = 4500 \]
Now multiply these two results:
\[ 10125 \times 4500 = 45656250 \]
Finally, take the square root:
\[ A = \sqrt{45656250} = 675 \text{ m}^2 \]

### Final Answer:
The area of the triangle is \(675 \text{ m}^2\).

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