Introduction

Relevance of the Topic

Triangles are elementary geometric figures that we constantly encounter in the world around us. Through knowledge about their sides and angles, we are able to understand most of the spatial phenomena that surround us. Classifying triangles according to their sides is crucial as it establishes the basis for the study of other topics in geometry, such as similarity, congruence, and the construction of more complex figures.

Contextualization

The classification of triangles is an integral part of the 1st year High School Mathematics curriculum, following the study of relationships and proportions in plane figures. The concepts and definitions established here will serve as a foundation for in-depth study of geometry throughout education. With the topic Triangles: Side Classification, we are continuing to develop students' ability to identify and apply basic geometric properties.

This topic fits perfectly along with others such as the classification of triangles by internal angles, the existence of Euclid's fifth proposition, and the exterior angle theorem. Moreover, the classification of triangle sides is directly and indirectly related to many other areas of mathematics, including Algebra (when studying the resolution of linear systems), Trigonometry (in the study of trigonometric ratios), and even Physics (in topics such as the resolution of forces into components). Therefore, it is essential that all students understand and master this subject.

Theoretical Development

Components

• Triangle: A plane figure formed by three line segments that intersect only at their ends, called vertices. The sum of the measures of any two sides of a triangle is always greater than the measure of the third side.
• Sides of a triangle: These are the line segments that connect the vertices of the triangle. Each triangle has three sides.
• Equality of sides: In the context of triangles, when the lengths of the sides are equal, the sides are called "congruent sides".
• Classification of triangles by sides: Triangles can be classified into three categories - Equilateral, Isosceles, and Scalene - according to the measure of their sides.

Key Terms

• Equilateral: Triangle with all congruent sides.
• Isosceles: Triangle with two congruent sides (of the same measure).
• Scalene: Triangle with all sides of different measures.

Examples and Cases

• Equilateral Triangle: All sides of an equilateral triangle measure the same amount. For example, a triangle with sides measuring 6 units each is an equilateral triangle.
• Isosceles Triangle: In an isosceles triangle, two of the sides have the same measure, while the third side has a different measure. For example, in a triangle with sides measuring 6, 6, and 8 units, the two 6-unit sides are congruent, and the third 8-unit side is different.
• Scalene Triangle: A scalene triangle has all sides with different measures. For example, a triangle with sides measuring 4, 5, and 7 units is a scalene triangle.

Detailed Summary

Relevant Points

• Definition of triangle: The triangle is a plane geometric figure formed by three line segments that meet in pairs, and the sum of the lengths of any two sides is always greater than the length of the third side.
• Types of triangles by the measure of the sides: Triangles can be classified into three main categories according to the measure of their sides - equilateral, isosceles, and scalene.
• Equilateral triangle: It is the one that has all three sides with the same measure. Its existence implies that all its internal angles have the same measure, 60 degrees.
• Isosceles triangle: It has two sides with the same measure. In the isosceles triangle, the angles opposite the congruent sides also have the same measure.
• Scalene triangle: It is the one in which all three sides have different measures. Consequently, the internal angles of a scalene triangle will also have different measures.

Conclusions

• The classification of triangles by sides is an essential tool in geometry, helping to identify and relate properties, such as the measure of internal angles.
• The classification of triangles by sides has important implications in other areas of mathematics, such as algebra and trigonometry. Each type of triangle has specific characteristics that can be used to solve problems in other disciplines.
• These basic concepts about triangles and their classifications will be the basis for the study of other topics in geometry, similarity and congruence of figures, among others.

Exercises

1. Determine if the following triangles are equilateral, isosceles, or scalene, giving a justification for each case:
   • Triangle ABC with sides AB = 5 cm, BC = 6 cm, and AC = 7 cm.
   • Triangle XYZ with sides XY = 3 m, YZ = 3 m, and XZ = 3 m.
   • Triangle MNP with sides MN = 4 cm, NP = 5 cm, and MP = 4 cm.
2. Draw an isosceles triangle where the measure of an internal angle is 80 degrees.
3. If the perimeter of an equilateral triangle is 24 cm, what is the measure of each of its sides?