Rectangular triangle: concept and properties

The solution of geometric problems requires a huge amount of knowledge. One of the fundamental definitions of this science is a right-angled 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 parties constituting a right angle bear the names of a leg, the third party, which is opposed to it, is called the hypotenuse.

If the legs in this figure are equal, it is called an isosceles right triangle. In this case, there is an accessory to two kinds of triangles, which means that the properties of both groups are respected. Recall that the angles at the base of an isosceles triangle are absolutely always equal, hence the sharp angles of such a figure will include 45 degrees.

The presence of one of the following properties allows us to state that one rectangular triangle is equal to the other:

1. The legs of two triangles are equal;
2. Figures have the same hypotenuse and one of the legs;
3. Equal to the hypotenuse and any of the acute angles;
4. The condition of equality of a leg and an acute angle is observed.

The area of a right-angled triangle can easily be computed with the help of standard formulas, and as a value equal to half the product of its legs.

In the right-angled triangle the following relations are observed:

1. The cathet is nothing but an average proportional hypotenuse and its projection on it;
2. If we 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 average proportional projection of the triangle's legs to its hypotenuse.

It is interesting that, whatever the right-angled triangle, these properties are always observed.

Pythagorean theorem

In addition to the above properties for rectangular triangles, the following condition is typical: the square of the hypotenuse is equal to the sum of the squares of the legs. This theorem is named after its founder - the theorem of Pythagoras. He discovered this relationship when he was studying the properties of squares constructed on the sides of a right-angled 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, we construct two squares. One side will have a hypotenuse, the other has 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 four triangles ABC and the second square, or as a side square, it is natural that these ratios will be equal. I.e:

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

With ² + 2 ab = a ² + b ² + 2 ab

As a result, we obtain: with ² = a ² + b ²

Thus, the geometric figure of a right-angled triangle corresponds not only to all 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 rectangular triangle, is found everywhere.