Rectangular triangle: concept and properties

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

By this concept is meant 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 the right angle are called the legs, the third side, which is opposite to it, is called the hypotenuse.

If the legs in such a figure are equal, it is called an isosceles right triangle. In this case, 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 always absolutely equal, therefore, the acute angles of such a figure will include 45 degrees.

The presence of one of the following properties allows us to state 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. hypotenuse and any of the acute angles are equal;
4. the condition of equality of the leg and the acute angle is observed.

The area of ​​a right-angled triangle is easily calculated using standard formulas, as well as a value equal to half the product of its legs.

The following relationships are observed in a right triangle:

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

It is interesting that no matter what a right triangle is, these properties are always respected.

Pythagorean theorem

In addition to the above properties, rectangular 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 called by the name of its founder - the Pythagorean theorem. He discovered this relationship when he studied the properties of squares built on the sides of a right triangle.

To prove the theorem, we construct a triangle ABC, the legs of which are denoted by a and b, and the hypotenuse c. Next, build two squares. One side will be 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 the four ABC triangles and the second square, or as the square of the side, naturally, these ratios will be equal. I.e:

with ² + 4 (ab / 2) = (a + b) ² , we transform the resulting expression:

s ² + 2 ab = a ² + b ² + 2 ab

As a result, we get: c ² = a ² + b ²

Thus, the geometric shape of a right 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 will be useful not only in science, but also in everyday life, since such a figure as a right-angled triangle is found everywhere.