The triangle is a three-sided polygon that gives rise to three vertices and three interior angles. It is the simplest figure, after the line in geometry. As a general rule, a triangle is represented by three capital letters of the vertices (ABC). Triangles are the most important geometric figures, since any polygon with a greater number of sides can be reduced to a succession of triangles, by drawing all the diagonals from a vertex, or by joining all their vertices with an interior point of the polygon.
It is important to note that among all the triangles the right triangle stands out, whose sides satisfy the metric relation known as Pythagorean theorem.
Herón de Alejandría was a Greek engineer and mathematician who lived during the 1st century BC, he wrote a work called La Métrica, where he devoted himself to the study of the volumes and areas of different surfaces and bodies. But undoubtedly the most important thing done by this mathematician was the well-known Heron's Formula, which is responsible for directly relating the area of a triangle with the lengths of its sides.
A right triangle consists of a 90 ° angle and two acute angles. Each acute angle of a right triangle has the functions of sine, cosine, and tangent. These, in turn, are points located on two of the three legs of a right triangle.
The sine of an angle is the ratio of the length of the opposite leg of the angle divided by the length of the hypotenuse.
The cosine of an angle is the ratio of the length of the leg adjacent to the angle divided by the length of the hypotenuse.
The tangent of an angle is the ratio of the length of the opposite leg of the angle divided by the length of the adjacent side of the angle.
Types of triangles
Table of Contents
The classification of triangles according to their sides and according to their angles is:
Triangles according to the length of their sides
According to the length of its sides, a triangle can be classified as equilateral, where the three sides of the triangle are equal; in isosceles, the triangle has two equal sides and one unequal, and in scalene, where the triangle has three unequal sides.
Equilateral triangle
This type of triangle has all three equal sides, that is, they are the same length. This type of triangle is widely used in practice, because its properties are symmetrical and easy to use.
Scalene triangle
This triangle has its three sides different from each other, that is, the lengths of its sides are different, they do not have any common side.
Isosceles triangle
It is the triangle whose two sides are equal, the third side is called the base. The angles in this base are reciprocally equal, if two angles of a triangle are equal, the sides opposite those angles will also be equal.
Triangles according to their angles
They can also be classified according to the measure of their angles, these can be:
Right triangle
If a triangle has a right angle or 90 ° angle, it is said to be a right angle. Another characteristic is that in the right triangle, the sides that form the right angle are called the legs and the opposite side is called the hypotenuse.
Obtuse triangle
It is the triangle that presents one of the three angles as obtuse; that is, an angle greater than 90 °.
Acute Triangle
It is the triangle where the three angles are acute; that is, angles less than 90 °.
Equiangular Triangle
These triangles are also called equilateral, their three internal sides are equal, with a measure of 60 ° each, and also, their three angles are congruent.
This triangle image has as its main characteristic that the sum of its three angles is always equal to 180 °. If we know two of them we can calculate how long the third will be.
The area of a triangle is equal to its base (any one of its sides) times its height (segment perpendicular to the base or to its extension, drawn from the vertex opposite the base side) divided by two, in other words, it is (base x height) / 2.
Through the following link //www.geogebra.org/m/BCA8uhHq you can see images of triangles according to their classification.
Elements of a triangle
Triangles have been analyzed in great detail since ancient civilizations. The Greek philosophers gave very detailed descriptions of its forms and elements, as well as their properties and their genuine relationships.
There are 5 elements of great interest in triangles that are:
Area of a triangle
The area of a triangle is the measure of the area enclosed by the three sides of the triangle. The classic formula for its calculation is: the measure of the base times the height and divided by two.
Median of a triangle
It is the segment established between the vertex and the midpoint of the opposite side. The medians of a triangle occurs at a point called the centroid or center of gravity of the triangle.
Mediatrix of a triangle
It is the line drawn perpendicular to the side at its midpoint. These occur at a point called the circumcenter, which is equidistant (is at the same distance) from the vertices of the same and is the center of a circle circumscribed to said triangle.
Bisector of a triangle
It is the interior ray of the angle that divides it into two equal angles. The bisectors of the interior angles coincide at a point called the incenter, which is equidistant from the sides of the triangle and is the center of a circle inscribed in it.
Height of a triangle
It is the perpendicular segment between the vertex and the opposite side. The three heights of a triangle meet at a point called the orthocenter.
Properties of a triangle
Each triangle verifies a very interesting set of essential geometric properties:
- Each side is smaller than the sum of the other two and greater than their difference.
- The three interior angles of a triangle always add a plane angle (180º). For this reason, equilateral triangles have three equal sides and three equal angles, with a value of 60º.
- The larger angle is opposite the longest side of the triangle and vice versa. Similarly, if two sides are equal, their opposite interior angles are also equal, and vice versa. In this case, for example, the equilateral triangles are regular.
Other definitions of triangle
Instrument triangle
The triangle presents another definition in the field of music, as a percussion instrument of indeterminate height, consisting of a metal bar bent in the shape of a triangle, open at one vertex, which is held with a finger or string, keeping it suspended in the air and is touched by hitting it with a metal rod. This instrument is very common in orchestras.
The sound of the triangle is of an indefinite height and sharp, for this reason it does not generate specific notes. The sound of this instrument will be open or closed as held by the musician. In addition, the triangle has a great sound, which allows it to be heard above the orchestra. This instrument measures approximately between 16 and 20 cm.
Hesselbach triangle
Hesselbach's triangle is a region located on the posterior wall of the inguinal region. This space is limited laterally by the inferior epigastric vessels (deep epigastric), below the inguinal ligament, and medially by the lateral border of the rectus abdominis muscle (anterior superior aspect of the abdomen).
An area is considered to be within the region, since it is a site where direct inguinal hernias are maintained. This ligament, the fascia and the inguinal trigone were discovered by the German surgeon Franz Kaspar Hesselbach, for this reason it was named the Hesselbach Triangle.
Love triangle
As defined above, a triangle is a geometric figure with three corners that converge and meet. The love triangle is not far from this definition. Basically it refers to a relationship of three, in which a man or a woman is romantically related to two people at the same time. In this situation you can arrive in a conscious and even unconscious way, which can make you love and hate yourself at the same time. Fundamentally, this depends on the corner you occupy in the triangle, which will also determine the ups and downs in your emotions or the enjoyment or not of this experience.
The human being is constantly looking for what he does not have, or what can be forbidden and unattainable. For example, he is always in search of complete happiness, of wanting everything, of owning everything, which is impossible, you never have everything in life.
In the field of astronomy; the triangle or Triangulum, is a small constellation of the Northern Hemisphere located between those of Andromeda, Pisces, Aries and Perseus.
