The perimeter of a triangle is 540 m and its sides are in the ratio 12 : 25 : 17. 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 1: Understand the problem
We know the perimeter of the triangle is 540 m and the sides are in the ratio 12:25:17.

### Step 2: Set up the equations
Let the sides of the triangle be represented as:
- Side a = 12x
- Side b = 25x
- Side c = 17x

The perimeter of the triangle can be expressed as:
\[ a + b + c = 540 \]
Substituting the expressions for a, b, and c:
\[ 12x + 25x + 17x = 540 \]

### Step 3: Combine like terms
Combine the terms on the left side:
\[ 54x = 540 \]

### Step 4: Solve for x
To find the value of x, divide both sides by 54:
\[ x = \frac{540}{54} = 10 \]

### Step 5: Find the lengths of the sides
Now that we have x, we can find the lengths of the sides:
- Side a = 12x = 12 * 10 = 120 m
- Side b = 25x = 25 * 10 = 250 m
- Side c = 17x = 17 * 10 = 170 m

### Step 6: Calculate the semi-perimeter (s)
The semi-perimeter s is given by:
\[ s = \frac{a + b + c}{2} = \frac{540}{2} = 270 \, m \]

### Step 7: Use Heron's formula to find the area
Heron's formula states that the area of the triangle can be calculated as:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we have:
\[ \text{Area} = \sqrt{270(270 - 120)(270 - 250)(270 - 170)} \]

### Step 8: Simplify the expression
Calculate each term:
- \( s - a = 270 - 120 = 150 \)
- \( s - b = 270 - 250 = 20 \)
- \( s - c = 270 - 170 = 100 \)

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

### Step 9: Calculate the area
Calculating the product:
\[ 270 \times 150 = 40500 \]
\[ 40500 \times 20 = 810000 \]
\[ 810000 \times 100 = 81000000 \]

Now, take the square root:
\[ \text{Area} = \sqrt{81000000} = 9000 \, m^2 \]

### Final Answer
The area of the triangle is \( 9000 \, m^2 \).

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