The Triangles Are a three-sided geometric figure called segments, whose junction forms the vertices that, in turn, form the three interior angles of the figure.
They are called properties to those characteristics that differentiate the geometric figures and that do not vary when projecting the figure from one plane to another, according to the investigations that began in century XVII, giving rise to the projective geometry.
Although there is no absolute certainty, it is believed that the first person to describe a triangle and make the respective geometric demonstrations using logical language was Thales of Miletus In the fifth century BC, approximately.
This statement could be true if one takes into account that Geometry, a science that studies the properties of geometric figures, was developed in the Ancient Egypt And in the Mesopotamian civilizations, from where it happened to the Greeks being the pioneers, Pythagoras And Euclid.
All the magnitudes that can be considered in a triangle (angles, sides, heights and medians), are called elements of a triangle. Studying these magnitudes is also called trigonometry.
The triangles were very useful when the first civilizations were launched to the study of the stars and to solve problems related to the construction, like the trisection of an angle, for example.
Main Properties of Triangles
Of the most notable properties of a triangle, stand out:
-The sum of the internal angles of a triangle always results in 180 °.
-In summing the lengths of two segments of a triangle, we always get a number greater than the length of the third side, and less than the difference.
An outer angle is equal to the sum of the two internal angles not adjacent to it.
-The triangles are always convex because none of their angles can exceed 180 °.
-A greater side always opposes the greater angle.
-In the triangles, the theorem of the breast is fulfilled:"The sides of a triangle are proportional to the sines of opposite angles."
- The cosine theorem is also fulfilled in a triangle and says:"The square of one side is equal to the sum of the squares of the other sides but twice the product of these sides by the cosine of the included angle."
-The middle base of a triangle measures the same as half of the parallel side.
- They are classified by the length of their sides or the amplitude of their angles.
-When a triangle has two equal sides, its opposite angles are also equal.
-A triangle is a rectangle (internal angle of 90 °) or an oblique angle (if none of its internal angles are straight or 90 °).
-The area of a triangle is equal to the result of multiplying the length of its base, by height, by two. This theory was demonstrated by Heron of Alexandria in the first book of a work that is attributed to him and that takes by name Metric (discovered in 1896).
-Any polygon can be divided into a finite number of triangles, this is achieved by triangulation.
-The perimeter of a triangle is equal to the sum of its three segments.
-Other theorem that is fulfilled in the triangles is the Pythagorean Theorem, according to which: a2 + b2 = c2; Where a and b are cathets and c is the hypotenuse.
-The triangles also have a measure of quality. The quality of a triangle (CT) results as a product: add the length of two sides and subtract the length of the third, dividing it by the product of its three sides. When CT = 1, we speak of an equilateral triangle; When CT = 0, there is a degenerate triangle; And when CT> 0.5 is about what is termed as a triangle of good quality.
-The congruence of triangles occurs when there is correspondence between the vertices of two triangles, so that the angle of the vertex and the sides that make up one of them, are congruent with those of the other triangle.
-Semblance of right triangles, is a property that is fulfilled when: they share the value of an acute angle; Share the same magnitude of two of their hicks; A catheter and the hypotenuse of one, are proportional to those of another.
"It is believed that Thales of Miletus relied on this law to calculate the height of an Egyptian pyramid and to determine the distance between a vessel and the coast.
Parts of a triangle
Side
The side of a triangle is the line connecting two vertices.
Vertex
It is the point of intersection of two segments.
Internal or internal angle
The inner angle is the opening level that is formed at the apex of a triangle.
Altitude
It is called altitude to the length of the straight line from a vertex to the diametrically opposite side.
Base
The base of the triangle depends on what altitude is being considered.
Half
It is a line that runs from the vertex towards the middle of the opposite side. Then, a triangle has three socks.
Bisector angle
It is thus called the line dividing an interior angle into two exactly alike. The length of this line can be known using the laws of Sine and Cosine.
Perpendicular bisector
It is a perpendicular line that crosses the midpoints of the segments of the triangle. When these lines join in the center of the triangle, they form the circle of the triangle whose midpoint is known as the circumcenter.
References
- Educar Chile (2010). All about the triangles. Recovered from: m.educarchile.cl
- The illustrated little Larousse (1999). Dictionary encyclopedic. Sixth edition. International co-edition.
- Geometrical figures (2014). History of geometry. Recovered from: m.figuras-geometricas8.webnode.es
- Mathematical Gacetilla (2001). Heron of Alexandria. Recovered from: mcj.arrakis.es
- Mathalino (s / f). Properties of a triangle. Retrieved from: mathalino.com.