Introduction

The Relevance of the Topic

The existence condition of the triangle, fundamental in Geometry, establishes the necessary requirements for three line segments to form a triangle. Understanding this concept is crucial, as it determines the boundaries of geometric formation, enabling the analysis and calculations of angles and sides of figures in this form. A failure in this perception hinders the understanding of many other subsequent concepts in geometry, thus limiting the ability to solve a wide range of mathematical problems.

Contextualization

The existence condition of the triangle is within the block of Plane Geometry in the 7th-grade mathematics curriculum. Previously, students have already been introduced to concepts such as point, line, plane, polygons, and angles. Next, they will delve into the classification of triangles, congruence and similarity of figures, as well as introduce the concept of triangle area. Therefore, understanding the necessary conditions for the formation of a triangle is a critical step for continuous learning and progress in the mathematical discipline.

Theoretical Development

Components

• Line Segments: It is the connection between two points. In the context of the triangle, we need to understand that the union of three line segments is the starting point for the formation of this figure. Without these segments, it is impossible to conceive a triangle.

• Lengths of the Triangle Sides: The three sides of the triangle are the line segments that compose it. The existence condition of the triangle establishes, among other conditions, that the sum of the lengths of any two sides must always be greater than the length of the third side. This is a fundamental pillar in defining a triangle.

Key Terms

• Triangle Existence Condition: Set of requirements that must be met for three line segments to form a triangle. These requirements involve the relationships between the lengths of the sides.

• Triangle: Flat geometric figure composed of three sides and three angles. It is the structure that arises when we join three line segments and is delimited when these segments are extended.

Examples and Cases

• Example 1: Suppose we have three line segments with lengths of 5 cm, 7 cm, and 10 cm. To verify if it is possible to form a triangle, we apply the existence condition. The sum of any two sides must be greater than the third. Checking: 5 + 7 = 12 (greater than 10), 5 + 10 = 15 (greater than 7), and 7 + 10 = 17 (greater than 5). Therefore, it is possible to form a triangle.

• Example 2: Now, consider three line segments with lengths of 3 cm, 4 cm, and 8 cm. Again, we apply the existence condition. Checking: 3 + 4 = 7 (less than 8). In this case, the existence condition is not met, so it is not possible to form a triangle with these line segments.

• Example 3: Finally, if we have three line segments with lengths of 6 cm, 8 cm, and 10 cm, the existence condition is met because the sum of any two sides is always greater than the length of the third side (6 + 8 = 14, 6 + 10 = 16, 8 + 10 = 18). In this case, it is possible to form a triangle.

Detailed Summary

Relevant Points

• Relevance of the Existence Condition: This condition is an indispensable requirement for three line segments to form a triangle. It establishes that the sum of the lengths of any two sides of the triangle must be greater than the length of the third side. Thus, understanding and mastering this condition are fundamental for the correct manipulation of triangles in mathematical exercises and problems.

• Line Segments and Length of Triangle Sides: Every triangle is formed from three line segments. The sum of the lengths of two of its sides will always be greater than the length of the third, one of the characteristics that define this geometric figure. A clear perception of this relationship is crucial to understand the existence condition of the triangle.

Conclusions

• Limiting Factor: Understanding the existence condition of the triangle is a limiting factor for future learning in Geometry. This condition is necessary for solving many problems involving triangles, including angle and side calculations and triangle classification.

• Practical Applications: The existence condition of the triangle is not an abstract concept; it has practical applications such as determining if three given lengths can form the sides of a triangular fence, or if a set of measurements can form a plane with triangular wings.

Exercises

1. Exercise 1: Given line segments of lengths 4 cm, 6 cm, and 8 cm, verify if it is possible to form a triangle by applying the existence condition.

2. Exercise 2: Using the existence condition of the triangle, check if the line segments of lengths 5 cm, 6 cm, and 14 cm can form a triangle.

3. Exercise 3: Find three consecutive integers whose sum is less than 28, and verify if these numbers can be the lengths of the sides of a triangle.