Perhaps the most basic, simple and interesting figure in geometry is a triangle. In a secondary school course, its basic properties are studied, but sometimes knowledge on this topic is formed incomplete. The types of triangles initially determine their properties. But this view remains mixed. Therefore, now we will analyze this topic in a little more detail.
Types of triangles depend on the degree measure of angles. These figures are acute, rectangular and obtuse. If all angles do not exceed 90 degrees, then the figure can safely be called acute-angled. If at least one angle of the triangle is 90 degrees, then you are dealing with a rectangular subspecies. Accordingly, in all other cases, the considered geometric figure is called obtuse-angled.
There are many tasks for acute subspecies. A distinctive feature is the internal location of the intersection points of the bisectors, medians and heights. In other cases, this condition may not be met. Determining the type of figure "triangle" is not difficult. It is enough to know, for example, the cosine of each angle. If any values are less than zero, then the triangle is obtuse in any case. In the case of a zero exponent, the figure hasright angle. All positive values are guaranteed to tell you that you have an acute-angled view.
One cannot but say about the right triangle. This is the most ideal view, where all the intersection points of medians, bisectors and heights coincide. The center of the inscribed and circumscribed circles also lies in the same place. To solve problems, you need to know only one side, since the angles are initially set for you, and the other two sides are known. That is, the figure is given by only one parameter. There are isosceles triangles. Their main feature is the equality of two sides and angles at the base.
Sometimes there is a question about whether there is a triangle with given sides. What you are really asking is whether this description fits the main species. For example, if the sum of two sides is less than the third, then in reality such a figure does not exist at all. If the assignment asks you to find the cosines of the angles of a triangle with sides 3, 5, 9, then there is an obvious catch. This can be explained without complex mathematical tricks. Suppose you want to get from point A to point B. The distance in a straight line is 9 kilometers. However, you remembered that you need to go to point C in the store. The distance from A to C is 3 kilometers, and from C to B - 5. Thus, it turns out that when moving through the store, you will walk one kilometer less. But since point C is not located on line AB, you will have to go an extra distance. Here a contradiction arises. This is, of course, a hypothetical explanation. Mathematics knows more than one way to prove thatall kinds of triangles obey the basic identity. It says that the sum of two sides is greater than the length of the third.
Any species has the following properties:
1) The sum of all angles equals 180 degrees.
2) There is always an orthocenter - the point of intersection of all three heights.
3) All three medians drawn from the vertices of interior corners intersect in one place.
4) A circle can be circumscribed around any triangle. You can also inscribe a circle so that it has only three points of contact and does not extend beyond the outer sides.
Now you are familiar with the basic properties that different types of triangles have. In the future, it is important to understand what you are dealing with when solving a problem.
