The sides of a triangular plot are in the ratio of 3 : 5 : 7 and its perimeter is 300 m, find its area :

Question

Accepted Answer

To find the area of the triangular plot with sides in the ratio 3:5:7 and a perimeter of 300 meters, we need to first determine the actual lengths of the sides.

1. **Determine the lengths of the sides:**
   - Let the sides of the triangle be $3x$, $5x$, and $7x$.
   - The perimeter of the triangle is given as 300 meters.
   - Therefore, $3x + 5x + 7x = 300$.
   - Simplifying, we get $15x = 300$.
   - Solving for $x$, we find $x = 20$.

Thus, the sides of the triangle are:
   - $3x = 60$ meters,
   - $5x = 100$ meters,
   - $7x = 140$ meters.

2. **Calculate the area using Heron's formula:**
   - First, calculate the semi-perimeter $s$ of the triangle:
     $$
     s = \frac{60 + 100 + 140}{2} = 150 \text{ meters}
     $$
   - Using Heron's formula, the area $A$ of the triangle is given by:
     $$
     A = \sqrt{s(s-a)(s-b)(s-c)}
     $$
     where $a = 60$, $b = 100$, and $c = 140$.

- Substitute the values into the formula:
     $$
     A = \sqrt{150(150-60)(150-100)(150-140)}
     $$
     $$
     A = \sqrt{150 \times 90 \times 50 \times 10}
     $$
     $$
     A = \sqrt{150 \times 45000}
     $$
     $$
     A = \sqrt{6750000}
     $$
     $$
     A = 1500\sqrt{3} \text{ m}^2
     $$

3. **Justification for Option 3:**
   - The calculated area of the triangle is $1500\sqrt{3} \text{ m}^2$, which matches Option 3.

4. **Explanation for why other options are incorrect:**
   - **Option 1: 150 m²** - This value is far too small given the dimensions of the triangle.
   - **Option 2: 900\sqrt{3} m²** - This value is incorrect as the calculation shows the area is larger.
   - **Option 4: 1500 m²** - This value does not include the $\sqrt{3}$ factor, which is essential in the correct calculation of the area using Heron's formula.

Therefore, the correct answer is indeed Option 3: $1500\sqrt{3} \text{ m}^2$.