Right triangle: concept and properties

2024 Author: Angel Austin | [email protected]. Last modified: 2023-12-17 05:14

Solving geometric problems requires a huge amount of knowledge. One of the fundamental definitions of this science is a right triangle.

This concept means a geometric figure consisting of three angles and

sides, and the value of one of the angles is 90 degrees. The sides that make up a right angle are called the leg, while the third side that is opposite it is called the hypotenuse.

If the legs in such a figure are equal, it is called an isosceles right triangle. In this case, there is an belonging to two types of triangles, which means that the properties of both groups are observed. Recall that the angles at the base of an isosceles triangle are absolutely always equal, therefore, the acute angles of such a figure will include 45 degrees each.

The presence of one of the following properties allows us to assert that one right triangle is equal to another:

1. the legs of two triangles are equal;
2. figures have the same hypotenuse and one of the legs;
3. the hypotenuse and anyfrom sharp corners;
4. the condition of equality of the leg and an acute angle is observed.

The area of a right triangle can be easily calculated both using standard formulas and as a value equal to half the product of its legs.

The following ratios are observed in a right triangle:

1. the leg is nothing but the mean proportional to the hypotenuse and its projection on it;
2. if you describe a circle around a right triangle, its center will be in the middle of the hypotenuse;
3. the height drawn from the right angle is the mean proportional to the projections of the legs of the triangle onto its hypotenuse.

It is interesting that no matter what the right triangle is, these properties are always observed.

Pythagorean theorem

In addition to the above properties, right triangles are characterized by the following condition: the square of the hypotenuse is equal to the sum of the squares of the legs.

This theorem is named after its founder - the Pythagorean theorem. He discovered this relation when he was studying the properties of squares built on the sides of a right triangle.

To prove the theorem, we construct a triangle ABC, whose legs we denote a and b, and the hypotenuse c. Next, we will build two squares. One side will be the hypotenuse, the other the sum of two legs.

Then the area of the first square can be found in two ways: as the sum of the areas of fourtriangles ABC and the second square, or as the square of the side, it is natural that these ratios will be equal. That is:

с² + 4 (ab/2)=(a + b)^{2, transform the resulting expression:}

c²+2 ab=a² + b^{2 + 2 ab}

As a result, we get: c²=a² + b²

Thus, the geometric figure of a right-angled triangle corresponds not only to all the properties characteristic of triangles. The presence of a right angle leads to the fact that the figure has other unique relationships. Their study is useful not only in science, but also in everyday life, since such a figure as a right-angled triangle is found everywhere.