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

Options

A. 375 `cm^(2)`
B. 750 `cm^(2)`
C. 250 `cm^(2)`
D. 500 `cm^(2)`

Answer

To solve the problem step by step, we will follow the same logic as in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Ratio of Sides**:
The sides of the triangle are in the ratio 5:12:13. Let's denote the sides as:
- Side A = 5k
- Side B = 12k
- Side C = 13k

2. **Finding the Perimeter**:
The perimeter of the triangle is given as 150 m. The perimeter (P) is the sum of all sides:
\[ P = A + B + C = 5k + 12k + 13k = 30k \]
Setting this equal to the given perimeter:
\[ 30k = 150 \]

3. **Calculating k**:
To find k, divide both sides by 30:
\[ k = \frac{150}{30} = 5 \]

4. **Calculating the Lengths of the Sides**:
Now, substitute k back into the expressions for the sides:
- Side A = \(5k = 5 \times 5 = 25\) m
- Side B = \(12k = 12 \times 5 = 60\) m
- Side C = \(13k = 13 \times 5 = 65\) m

5. **Calculating the Semi-Perimeter (s)**:
The semi-perimeter (s) is given by:
\[ s = \frac{A + B + C}{2} = \frac{150}{2} = 75 \text{ m} \]

6. **Using Heron's Formula to Find the Area**:
Heron's formula for the area (A) of a triangle is:
\[ A = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values we found:
\[ A = \sqrt{75(75 - 25)(75 - 60)(75 - 65)} \]
Simplifying the terms:
\[ A = \sqrt{75 \times 50 \times 15 \times 10} \]

7. **Calculating the Area**:
Now we calculate the area step by step:
- First, calculate \(75 \times 50 = 3750\)
- Then, \(15 \times 10 = 150\)
- Now multiply these results: \(3750 \times 150 = 562500\)
- Finally, take the square root:
\[ A = \sqrt{562500} = 750 \text{ m}^2 \]

### Final Answer:
The area of the triangle is **750 m²**.