Determining the Length of the Third Side of a Triangle using Triangle Inequality Theorem

In the field of geometry, the triangle inequality theorem plays a crucial role in determining the possible lengths of the sides of a triangle. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. By applying this theorem, we can explore the constraints on the third side of a triangle when two sides are given. This article will illustrate how to apply the triangle inequality theorem to determine the range of possible values for the third side of a triangle.

Introduction

The triangle inequality theorem is a fundamental concept in geometry that helps us understand the geometric constraints that must be satisfied for a set of three line segments to form a triangle. It is especially useful in problems where we need to determine the maximum or minimum possible lengths of the sides of a triangle based on the given measurements.

Triangle Inequality Theorem

The triangle inequality theorem can be stated as follows: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Mathematically, for a triangle with sides of lengths (a), (b), and (c), the following conditions must be satisfied:

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

Applying the Theorem to a Triangle with Given Sides

Consider a triangle with two known sides of lengths 5 cm and 1.5 cm. Let the third side be denoted as (c). To determine the possible lengths of (c), we apply the triangle inequality theorem to each pair of sides.

Step-by-Step Application of the Theorem

Step 1: Applying (a b > c)

In this case, (a 5 text{ cm}) and (b 1.5 text{ cm}), so we have: [5 1.5 > c implies 6.5 > c implies c

Step 2: Applying (a c > b)

This condition simplifies to: [5 c > 1.5 implies c > 1.5 - 5 implies c > -3.5]

Since side lengths must be positive, this condition is always true.

Step 3: Applying (b c > a)

This condition simplifies to: [1.5 c > 5 implies c > 5 - 1.5 implies c > 3.5]

Combining the results from Steps 1, 2, and 3, we find that the length of the third side (c) must satisfy:

[3.5

Conclusion

Thus, the length of the third side of the triangle cannot be less than or equal to 3.5 cm or greater than or equal to 6.5 cm. Specifically, it cannot be:

[c leq 3.5 text{ cm or } c geq 6.5 text{ cm}]

This example illustrates the importance of the triangle inequality theorem in determining the possible lengths of the sides of a triangle based on the given measurements.

Additional Examples

Example 1:

Given two sides of lengths 5 cm and 1.5 cm, the length of the third side, (c), must satisfy:

[3.5

Example 2:

Consider a triangle where one side is 5 cm and the other is 9 cm. The third side, (c), must satisfy:

[5 c > 9 implies c > 4] [c 5 > 9 implies c > 4] [c 9 > 5 implies c > -4 quad (text{always true})]

Additionally:

[5 9 > c implies c

Therefore, the length of the third side must be:

Example 3:

For a triangle with sides 4 cm and 1.5 cm, the length of the third side, (c), must satisfy:

[4 1.5 > c implies c 1.5 implies c > -2.5 quad (text{always true})] [1.5 c > 4 implies c > 2.5]