Geometry is one of the fundamental pillars of mathematics and one of the most important basic concepts within it is the triangleA triangle is a geometric figure consisting of three sides, three vertices and three angles. Although it may seem simple, the study of the different types of triangles reveals a world rich in properties and characteristics that have practical applications in many disciplines, such as physics, engineering and architecture.
In this article, we will delve into the types of triangles according to its sides and angles, and we will explore advanced concepts such as straight lines and notable points in a triangle. These are essential elements to understand its geometric behavior and associated phenomena.
What is a Triangle?
A triangle is a three-sided polygon that also has three angles and three vertices. It is one of the most basic and fundamental shapes in geometry. The sides are line segments that meet at points called vertices, and each internal angle is the space formed between two sides. The sum of the three internal angles of any triangle is always 180 degrees, which is a key characteristic of this shape.
Types of Triangles According to Their Sides
The classification of triangles begins with the length of their sides. Depending on the lengths of their three sides, triangles can be divided into three categories:
- Equilateral triangle: In an equilateral triangle, all three sides have the same length. As a result, its three internal angles are equal, each measuring 60°. Being a completely symmetrical figure, the equilateral triangle is a special case of regular polygons. Furthermore, these characteristics make it an ideal figure in certain architectural and structural applications due to its stability and symmetry.
- Isosceles triangle: In this type of triangle, two sides have the same length, while the third side is different. This equal length of two sides also determines that the angles opposite these sides are equal. Isosceles triangles are commonly used in contexts where some symmetry is necessary, either in mathematical analysis or in design and construction applications.
- Scalene triangle: This type of triangle is characterized by having three sides of different lengths. Because of this, none of the internal angles has the same measurement. Scalene triangles are geometrically asymmetrical and are essential for studying non-symmetrical and more complex situations in mathematics and trigonometry.
Types of Triangles According to Their Angles
Now we move on to classifying triangles according to their interior angles. Triangles can have different combinations of angles:
- Acute triangle: In acute triangles, the three internal angles are less than 90ºIn other words, they are acute angles. This type of triangle is considered more “closed” in its arrangement and is used in the analysis of compact shapes in geometry and design.
- Right triangle: The right triangle is one of the best known and most studied due to its relationship with the famous Pythagorean theorem. A right triangle has a right angle, that is, an angle that measures exactly 90°. In this case, the sides that form the right angle are called peccaries, while the opposite side is called hypotenuseThis type of triangle has important practical applications in construction, navigation, and in the science of applied trigonometry.
- Obtuse triangle: An obtuse triangle has one internal angle that is greater than 90°, making it a more "open" type of triangle than the others. This type of triangle is used to model situations that require a greater amplitude in its angular structure.
Notable Lines and Notable Points in a Triangle
Within the geometry of triangles it is not only important to classify by their sides and angles. remarkable straight lines They also play a crucial role in geometric analysis, as they join special points in triangles and intersect at specific locations, called notable pointsThere are four main notable lines:
- Medium: Medians are straight lines that join a vertex of the triangle with the midpoint of its opposite side. The point where these medians intersect is called center of gravity. This is the center of mass of the triangle, a point of geometric balance.
- Bisectors: They are perpendicular lines that pass through the midpoint of each side of the triangle. The point where the three medians intersect is called circumcenter, and is the center of the circumscribed circle that passes through the three vertices of the triangle.
- Heights: These are the straight lines that start from a vertex and fall perpendicularly on the opposite side, or its extension. The point where the three heights meet is called orthocenter.
- Bisectors: They are straight lines that bisect, or divide into two equal parts, each angle of the triangle. The point where these bisectors meet is the incenter, which is the center of the circle inscribed within the triangle.
Mathematical Properties of Triangles
One of the most outstanding geometric properties of triangles is that the The sum of its internal angles is always 180º, regardless of the type of triangle. This property is essential for solving geometric problems that involve determining angles and sides in different contexts.
El Pythagoras theorem is one of the most important mathematical properties related to right triangles. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs:
c² = a² + b²
This relationship is vital in applications that require the calculation of distances, such as in engineering and physics.
To calculate the area of a triangle, the most common formula can be used:
Area = (base * height) / 2
However, if the height is not known, the area can also be calculated using the Heron's formula, which determines the area based on the lengths of the three sides:
Area = √[s * (s – a) * (s – b) * (s – c)]
where 's' is the semiperimeter of the triangle calculated as:
s = (a + b + c) / 2
Applications of Triangles in Real Life
Triangles are not only important geometric figures in mathematics, but they also have numerous practical uses in the world around us. A typical example is in architecture, where triangles are used for their inherent stability. Trusses and triangular structures, which were already used in ancient construction, provide an efficient way of distributing weight on roofs, bridges and tall buildings. The reason for this is that a triangle is the only geometric figure that cannot be altered without changing the lengths of its sides.
En structural engineeringTriangles are used in suspension bridges, where loads are distributed between cables arranged in a triangular shape, thus optimizing strength and weight distribution.
En astronomy y navigation, the principle of triangulation is crucial for determining exact positions. By measuring distances from two known points, the location of a third point can be calculated using the properties of triangles.
Even in everyday objects, such as traffic signs or some furniture, triangles play an important role by offering stability and balance with an efficient use of materials.
The triangle is not just a figure visible in textbooks. Its use is present in modern technology and ancient structures, showing its durability and versatility throughout the ages.