The area of a triangle is `150 cm^(2)` and its sides are in the ratio `3:4:5`. What is its perimeter.

Options

A. `10 cm`
B. `30 cm`
C. `45 cm`
D. `60cm`

Answer

To solve the problem, we need to find the perimeter of a triangle whose area is given as \(150 \, \text{cm}^2\) and whose sides are in the ratio \(3:4:5\).

### Step-by-Step Solution:

1. **Understanding the Ratio of the Sides:**
The sides of the triangle can be expressed in terms of a common variable \(k\):
- Let the sides be \(3k\), \(4k\), and \(5k\).

2. **Calculating the Semi-Perimeter:**
The semi-perimeter \(s\) of the triangle is given by:
\[ s = \frac{3k + 4k + 5k}{2} = \frac{12k}{2} = 6k \]

3. **Using Heron's Formula for Area:**
Heron's formula states that the area \(A\) of a triangle can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \(a\), \(b\), and \(c\) are the sides of the triangle. Substituting the values:
\[ A = 150 \, \text{cm}^2 \]
\[ s = 6k, \quad a = 3k, \quad b = 4k, \quad c = 5k \]
Now substituting into Heron's formula:
\[ 150 = \sqrt{6k(6k - 3k)(6k - 4k)(6k - 5k)} \]
Simplifying the terms inside the square root:
\[ 150 = \sqrt{6k(3k)(2k)(k)} = \sqrt{36k^4} = 6k^2 \]

4. **Solving for \(k\):**
Now, we can equate and solve for \(k\):
\[ 150 = 6k^2 \]
Dividing both sides by 6:
\[ k^2 = 25 \]
Taking the square root:
\[ k = 5 \]

5. **Finding the Length of the Sides:**
Now we can find the lengths of the sides:
- \(a = 3k = 3 \times 5 = 15 \, \text{cm}\)
- \(b = 4k = 4 \times 5 = 20 \, \text{cm}\)
- \(c = 5k = 5 \times 5 = 25 \, \text{cm}\)

6. **Calculating the Perimeter:**
The perimeter \(P\) of the triangle is given by:
\[ P = a + b + c = 15 + 20 + 25 = 60 \, \text{cm} \]

### Final Answer:
The perimeter of the triangle is \(60 \, \text{cm}\).