An important concept in geometry, as a science, is the similarity of figures. Knowing this property allows you to solve a huge number of problems, including in real life.
The concepts
Such figures are those that can be converted into each other by multiplying all sides by a certain factor. In this case, the corresponding angles should be equal.
Let us consider in more detail the signs of the similarity of triangles. In total, there are three rules that allow us to argue that such figures have this property.
The first sign of the similarity of triangles requires that there is an equality of two pairs of corresponding angles.
According to the second rule, the figures in question are considered similar when two sides of one are proportional to the corresponding segments of the other. Moreover, the angles that are formed by them must be equal.
And finally, the third sign: triangles are similar if all their sides are respectively proportional.
There are such figures, which according to some properties can be attributed to special types (equilateral, isosceles, rectangular). To state that such triangles are similar, fewer conditions are necessary. For example, we will consider the similarity signs of rectangular
triangles:
- the hypotenuse and one of the legs of one are proportional to the corresponding sides of the other;
- any acute angle of one figure is equal to the same in another.
If the signs of similarity of triangles are observed, the following properties take place:
- the ratio of their linear elements (medians, bisectors, heights, perimeters) is equal to the similarity coefficient;
- if you find the result of dividing the areas, we get the square of this number.
Application
The properties considered allow us to solve a huge number of geometric problems. They are widely used in life. Knowing the signs of the similarity of triangles, you can determine the height of an object or calculate the distance to an inaccessible point.
In order to find out, for example, the height of a tree, a pole is mounted strictly vertically at a predetermined distance, on which a rotating plank is fixed. It is oriented to the top of the subject and marks the point on the ground where the line continuing it crosses the horizontal surface. We get similar right-angled triangles. By measuring the distance from the point to the pole, and then to the subject, we find the similarity coefficient. Knowing the height of the pole, you can easily calculate the same parameter for a tree.
To find the distance between two points on the ground, choose one more on the plane. Then we measure the distance from it to the accessible. Connect all the points on the ground and measure the angles that are adjacent to the known side. Having built a similar triangle on paper and determining the aspect ratio of the two figures, we can easily calculate the distance between the points.
Thus, the similarity signs 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.