The perimeter of a triangle is 300 m and its sides are in the ratio 3:5:7. Find its area.

Options

A. `1800 sqrt3 m^(2)`
B. `1500 sqrt3 m^(2)`
C. `4500 m^(2)`
D. `2500 m^(2)`

Answer

To solve the problem step-by-step, we need to follow these steps:

### Step 1: Understand the given information
The perimeter of the triangle is given as 300 m, and the sides are in the ratio of 3:5:7.

### Step 2: Express the sides in terms of a variable
Let the sides of the triangle be represented as:
- First side = 3r
- Second side = 5r
- Third side = 7r

Where \( r \) is a common multiple.

### Step 3: Write the equation for the perimeter
The perimeter of the triangle can be expressed as:
\[ 3r + 5r + 7r = 300 \]

### Step 4: Simplify the equation
Combine the terms:
\[ 15r = 300 \]

### Step 5: Solve for \( r \)
To find the value of \( r \), divide both sides by 15:
\[ r = \frac{300}{15} = 20 \]

### Step 6: Calculate the lengths of the sides
Now substitute \( r \) back into the expressions for the sides:
- First side = \( 3r = 3 \times 20 = 60 \) m
- Second side = \( 5r = 5 \times 20 = 100 \) m
- Third side = \( 7r = 7 \times 20 = 140 \) m

### Step 7: Calculate the semi-perimeter
The semi-perimeter \( s \) is given by:
\[ s = \frac{\text{Perimeter}}{2} = \frac{300}{2} = 150 \text{ m} \]

### Step 8: Use Heron's formula to find the 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 \( a, b, c \) are the lengths of the sides.

Substituting the values:
- \( a = 60 \) m
- \( b = 100 \) m
- \( c = 140 \) m

Now calculate:
\[ A = \sqrt{150(150-60)(150-100)(150-140)} \]
\[ = \sqrt{150 \times 90 \times 50 \times 10} \]

### Step 9: Simplify the expression
Calculate the product inside the square root:
\[ = \sqrt{150 \times 90 \times 50 \times 10} \]
\[ = \sqrt{150 \times 45000} \]
\[ = \sqrt{6750000} \]

### Step 10: Final calculation
To simplify further:
\[ = 150 \sqrt{300} \text{ m}^2 \]

Thus, the area of the triangle is:
\[ A = 1500 \sqrt{3} \text{ m}^2 \]