Step 1: Determine the Lengths of the Sides
To find the lengths of the sides of the triangle, we first express the sides in terms of a variable based on their ratio:
- Let the sides be represented as:
- Side 1 = 3x
- Side 2 = 5x
- Side 3 = 7x
Next, we know the perimeter of the triangle is the sum of all sides:
- Perimeter = 3x + 5x + 7x = 15x
Given that the perimeter is 300 m:
- 15x = 300
- Therefore, x = 20
Now we can find the lengths of the sides:
- Side 1 = 3x = 60 m
- Side 2 = 5x = 100 m
- Side 3 = 7x = 140 m
Step 2: Calculate the Area using Heron's Formula
To find the area, we can use Heron’s formula, which requires the semi-perimeter (s):
- Semi-perimeter (s) = Perimeter / 2 = 300 / 2 = 150 m
Now, we can use Heron’s formula:
- Area = sqrt[s(s - a)(s - b)(s - c)]
Where:
- a = 60 m
- b = 100 m
- c = 140 m
Calculating each term:
- s - a = 150 - 60 = 90
- s - b = 150 - 100 = 50
- s - c = 150 - 140 = 10
Now substituting these values into the formula:
- Area = sqrt[150 * 90 * 50 * 10]
Final Calculation
- Area = sqrt[7500000] = 2738.61 m² (approximately)
Thus, the area of the triangle is approximately 2738.61 square meters.