Triangles. Properties and Pythagorean Theorem

What is a triangle?

Triangles are geometric figures with three sides and three angles. They are one of the most basic and fundamental shapes in Euclidean geometry. Triangles are classified according to their sides and angles.

Classification of triangles according to their sides: equilateral, isosceles and scalene triangles

In terms of sides, triangles can be equilateral, isosceles or scalene. An equilateral triangle has three equal sides, while an isosceles triangle has two equal sides and one different side. Finally, a scalene triangle has three different sides.

Classification of triangles according to their angles: acute, obtuse and right triangles

As for the angles, triangles can be acute, obtuse or right triangles. An acute triangle has all three acute angles, that is, less than 90 degrees. An obtuse triangle has one obtuse angle, that is, greater than 90 degrees. And a right triangle has a right angle of 90 degrees.

Properties of triangles

Triangles have unique properties and characteristics. The sum of the internal angles of a triangle is always equal to 180 degrees, which is one of the fundamental bases of geometry. In addition, any side of a triangle is always less than the sum of the other two sides, a property known as the triangular inequality, which ensures the viability of its shape. These properties are essential for understanding how triangles interact within more complex geometric figures, as well as in practical applications, such as the design of structures and mechanical systems where stability and precision are required.

Properties of right triangles. Pythagorean Theorem

The Pythagorean theorem is a fundamental rule in right triangles. It states that the square of the hypotenuse, which is the side opposite the 90 degree angle, equals the sum of the squares of the other two sides. Rectangular triangles are fundamental in trigonometry, where ratios such as sine, cosine and tangent are used to relate the sides and angles of these geometric figures.

Applications of triangles

In addition to their importance in Euclidean geometry, triangles are widely used in various areas, such as architecture, engineering and physics. Their simple and versatile structure makes them key elements in the construction of stable structures and in the description of natural and physical phenomena. In architecture, triangles are used to create geometric designs that distribute weight evenly and ensure the stability of buildings such as bridges and towers. In engineering, the properties of triangles are used to design mechanical and structural systems that support forces and movements. In physics, on the other hand, triangles are essential for modeling force dynamics, such as projectile trajectories and energy transmission. They also represent a fundamental component in the study of trigonometry, which is applied in fields as varied as navigation, astronomy and computational simulations.