Calculating the Area of a Triangle with a Given Ratio and Perimeter

In this article, we explore how to calculate the area of a triangle given that its sides are in a specific ratio and its perimeter is known. We will discuss two examples and delve into the step-by-step process, highlighting the importance of checking if the given sides can form a triangle, and using Heron's formula for accurate area calculation.

Example 1: Perimeter 100 cm, Side Ratio 1:2:3

Consider a triangle with a perimeter of 100 cm, and sides in the ratio 1:2:3. To find the area of this triangle, follow these steps:

Determine the lengths of the sides

Let the sides of the triangle be x, 2x, 3x. The perimeter is given by: x 2x 3x 100. This simplifies to: 6x 100. Solving for x, we get: x 100 / 6 ≈ 16.67 cm. Therefore, the lengths of the sides are approximately: x 16.67 cm 2x 33.33 cm 3x 50 cm

Check if these sides can form a triangle

For three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Checking the inequalities: 16.67 33.33 50 (not greater than 50) 16.67 50 66.67 33.33 33.33 50 83.33 16.67 Since one of the inequalities is not satisfied, these lengths cannot form a triangle. Therefore, the area of the triangle is 0 cm2.

Example 2: Perimeter 180 cm, Side Ratio 2:3:4

Now, consider a triangle with a perimeter of 180 cm, and sides in the ratio 2:3:4. To find the area of this triangle, follow these steps:

Determine the lengths of the sides

Given the ratio, the sides can be represented as: 2x, 3x, 4x. The perimeter is given by: 2x 3x 4x 180. This simplifies to: 9x 180. Solving for x, we get: x 180 / 9 20. Therefore, the lengths of the sides are: a 2x 40 cm b 3x 60 cm c 4x 80 cm

Calculate the semi-perimeter, s

s (a b c) / 2 (40 60 80) / 2 90 cm

Use Heron's formula to find the area

Heron's formula for the area of a triangle is: A sqrt{s(s - a)(s - b)(s - c)}. Substituting the values: A sqrt{90(90 - 40)(90 - 60)(90 - 80)} A sqrt{90 * 50 * 30 * 10} A sqrt{1350000} A ≈ 1161.89 cm2

Conclusion

Calculating the area of a triangle based on given ratios and perimeter involves both algebraic manipulation and geometric principles. The process includes checking for the triangle inequality and using Heron's formula for accurate calculations. Understanding these steps is crucial in various mathematical and real-world applications, such as land surveying and construction.