Exploring the Triangular Inequality: A Deep Dive into Side Lengths and Perimeter

In the study of geometry, a fundamental concept is the triangular inequality, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. This theorem ensures that a triangle can be physically realized with the given side lengths. The provided scenario presents a common misconception regarding the application of triangular inequality in the context of perimeter calculation. This article delves into this issue, clarifying the underlying principles and providing insight into solving related problems.

Understanding the Triangular Inequality

The triangular inequality is a critical principle in geometry. It asserts that for any triangle with sides of lengths (a), (b), and (c), the following conditions must be met:

(a b > c) (a c > b) (b c > a)

These conditions ensure that the sides can form a triangle. If any of these conditions are not satisfied, the sides cannot form a triangle, and the structure would collapse into a degenerate shape or non-existent form.

The Given Problem and Its Resolution

You presented a problem where the two sides of a triangle are (85 , text{cm}) and (20 , text{cm}), and the perimeter of the triangle is (48 , text{cm}). The question arises as to how a triangle can have one side as 85 cm and a total perimeter of only 48 cm.

Let's denote the sides of the triangle as a 85 cm, b 20 cm, and c as the unknown third side. The perimeter P is given by:

[P a b c]

Substituting the known values:

[48 85 20 c]

By simplifying this equation:

[48 105 c]

Solving for c:

[c 48 - 105 -57]

As a negative side length is nonsensical in this context, we can conclude that a triangle with these side lengths cannot exist. The negative value indicating that these sides violate the triangular inequality principle. This means that a triangle with one side as 85 cm and another side as 20 cm cannot have a total perimeter of 48 cm.

Revising the Perimeter Calculation

To correctly calculate the side lengths given the perimeter, you need to ensure that the triangular inequality holds. Let's consider the correct scenario where the perimeter is 85 cm and the two sides are 48 cm and 20 cm:

[P a b c 85]

Given a 48 cm and b 20 cm, we solve for the third side c:

[85 48 20 c]

By substituting and simplifying:

[85 68 c]

Solving for c:

[c 85 - 68 17]

With the third side of 17 cm, the corrected perimeter is indeed 85 cm, and all triangular inequality conditions are satisfied:

(48 20 > 17) (true) (48 17 > 20) (true) (20 17 > 48) (true)

This scenario is consistent with the rules of geometry and can form a valid triangle.

Conclusion and Final Thoughts

The provided problem and the attempted solution highlight the importance of understanding and applying the triangular inequality theorem correctly. A triangle cannot have one side as 85 cm, another as 20 cm, and a perimeter of 48 cm, as this violates the fundamental principles of geometry. By ensuring that the sum of any two sides is greater than the third side, the challenge of determining side lengths given the perimeter can be accurately resolved. In the correct scenario (perimeter 85 cm, sides 48 cm and 20 cm), the third side is 17 cm, providing a valid and geometrically sound configuration.

Understanding and applying these concepts ensures accurate and meaningful geometric analysis. This principle is not only crucial for academic and practical geometry applications but also underpins many real-world scenarios where the properties of triangles are essential.