Substituting the values into the formula gives √(150 × 90 × 50 × 10). We can factor this as √((150 × 50) × (90 × 10)) = √(7500 × 900) = √(2500 × 3 × 900) = 50 × 30√3 = 1500√3 m².

Substituting the values into the formula gives √(150 × 90 × 50 × 10). We can factor this as √((150 × 50) × (90 × 10)) = √(7500 × 900) = √(2500 × 3 × 900) = 50 × 30√3 = 1500√3 m².

The side of a triangular plot are in the ratio 3:5:7,it's perimeter is 300 m. Find its area

Answer: $1500\sqrt{3}\text{ m}^2$

Step-by-step solution

Step 1: Find the side lengths using the perimeter

Let the side lengths of the triangular plot be $a = 3x$ , $b = 5x$ , and $c = 7x$ in meters. Since the perimeter is the sum of all three sides, we set their sum equal to $300\text{ m}$ . Solving this equation gives $x = 20$ , which yields side lengths of $60\text{ m}$ , $100\text{ m}$ , and $140\text{ m}$ .

Step 2: Calculate the semi-perimeter

The semi-perimeter $s$ is defined as half of the perimeter of the triangle. Dividing the given perimeter $300\text{ m}$ by $2$ gives $s = 150\text{ m}$ .

Step 3: Apply Heron's formula

We use Heron's formula to calculate the area of the triangle from its side lengths and semi-perimeter. We first compute the differences: $s - a = 150 - 60 = 90\text{ m}$ , $s - b = 150 - 100 = 50\text{ m}$ , and $s - c = 150 - 140 = 10 weeks \implies 10\text{ m}$ .

Step 4: Compute the final area

Substituting the values into the formula gives $\sqrt{150 \times 90 \times 50 \times 10}$ . We can factor this as $\sqrt{(150 \times 50) \times (90 \times 10)} = \sqrt{7500 \times 900} = \sqrt{2500 \times 3 \times 900} = 50 \times 30\sqrt{3} = 1500\sqrt{3}\text{ m}^2$ .

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