Signs of the similarity of triangles: concepts and scope

An important concept in geometry, as a science, is the resemblance of figures. Knowledge of this property allows solving a huge number of tasks, including in real life.

Concepts

Such figures are those that can be translated into each other by multiplying all sides by a certain coefficient. The corresponding angles must be equal.

Let us consider in more detail the similarity of triangles. In total, there are three rules that allow us to assert that such figures have this property.

The first sign of the similarity of triangles requires that the equality of two pairs of corresponding angles take place.

According to the second rule, the figures considered are considered similar when two sides of one are proportional to the corresponding segments of the other. In this case, the angles that are formed by them must be equal.

And, finally, the third sign: triangles are similar if all their sides are proportionally proportionate.

There are such figures, which for some properties can be classified as special types (equilateral, isosceles, rectangular). To argue that such triangles are similar, it is necessary to perform fewer conditions. For example, we consider the signs of the similarity of rectangular Triangles:

1. The hypotenuse and one of the legs of one are proportional to the corresponding sides of the other;
2. Any acute angle of one figure is equal to the same in the other.

If the signs of similarity of triangles are observed, the following properties hold:

1. The ratio of their linear elements (medians, bisectors, heights, perimeters) is equal to the similarity coefficient;
2. If we find the result of dividing the areas, we obtain the square of this number.

Application

The properties considered allow solving a huge number of geometric problems. They are widely used in life. Knowing the signs of similarity of triangles, you can determine the height of an object or calculate the distance to an inaccessible point.

To find out, for example, the height of a tree, at a predetermined distance, a pole is fixed strictly vertically, on which a rotating bar is fixed. It is oriented to the top of the object and marks on the ground a point where the line continuing it will cross the horizontal surface. We get similar rectangular triangles. Measuring the distance from the point to the pole, and then to the object, we find the similarity coefficient. Knowing the height of the pole, you can easily calculate the same parameter for the tree.

To find the distance between two points on the terrain, we select on the plane one more. Then we measure the distance from it to the available one. We will connect all the points on the terrain and measure the angles that are adjacent to the known side. Having built a similar triangle on paper and determining the ratio of the sides of the two figures, we easily calculate the distance between the points.

Thus, the signs of the similarity of triangles are one of the most important concepts of geometry. It is widely used not only for scientific purposes, but also for other needs.