The sides of a triangle are in the ratio 5 : 12 : 13 and its perimeter is 150 m. Find the area of the triangle.

Options

A. 450 `m^(2)`
B. 750 `m^(2)`
C. 650 `m^(2)`
D. 550 `m^(2)`

Answer

To find the area of the triangle with sides in the ratio 5:12:13 and a perimeter of 150 m, we can follow these steps:

### Step 1: Determine the sides of the triangle
Given the ratio of the sides is 5:12:13, we can express the sides in terms of a variable \( x \):
- Side 1 (a) = \( 5x \)
- Side 2 (b) = \( 12x \)
- Side 3 (c) = \( 13x \)

### Step 2: Set up the equation for the perimeter
The perimeter of the triangle is the sum of its sides:
\[ 5x + 12x + 13x = 150 \]
Combining the terms gives:
\[ 30x = 150 \]

### Step 3: Solve for \( x \)
To find \( x \), divide both sides of the equation by 30:
\[ x = \frac{150}{30} = 5 \]

### Step 4: Calculate the lengths of the sides
Now substitute \( x \) back into the expressions for the sides:
- Side 1 (a) = \( 5x = 5 \times 5 = 25 \) m
- Side 2 (b) = \( 12x = 12 \times 5 = 60 \) m
- Side 3 (c) = \( 13x = 13 \times 5 = 65 \) m

### Step 5: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is given by:
\[ s = \frac{a + b + c}{2} = \frac{25 + 60 + 65}{2} = \frac{150}{2} = 75 \text{ m} \]

### Step 6: Use Heron's formula to find the area
Heron's formula states:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we found:
\[ \text{Area} = \sqrt{75(75-25)(75-60)(75-65)} \]
Calculating each term:
- \( s - a = 75 - 25 = 50 \)
- \( s - b = 75 - 60 = 15 \)
- \( s - c = 75 - 65 = 10 \)

So, we have:
\[ \text{Area} = \sqrt{75 \times 50 \times 15 \times 10} \]

### Step 7: Simplify the expression
Calculating the product inside the square root:
\[ 75 \times 50 = 3750 \]
\[ 3750 \times 15 = 56250 \]
\[ 56250 \times 10 = 562500 \]
Thus, we have:
\[ \text{Area} = \sqrt{562500} \]

### Step 8: Calculate the square root
Finding the square root:
\[ \sqrt{562500} = 750 \]

### Final Answer
The area of the triangle is \( 750 \, \text{m}^2 \).

---