Perhaps the most basic, simple and interesting figure in geometry is a triangle. A high school course studies its basic properties, but sometimes knowledge on this topic is formed incomplete. Types of triangles initially determine their properties. But this view remains mixed. Therefore, we will now analyze this topic in more detail.
The types of triangles depend on the degree measure of angles. These figures are acute, rectangular and obtuse. If all angles do not exceed the value of 90 degrees, then the figure can safely be called acute-angled. If at least one corner 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.
There are many tasks for acute-angle subspecies. A distinctive feature is the internal location of the intersection points of bisectors, medians and heights. In other cases, this condition may not be fulfilled. It is not difficult to determine the type of figure “triangle”. It is enough to know, for example, the cosine of each angle. If any values are less than zero, then the triangle is in any case obtuse. In the case of a zero indicator, the figure has a right angle. All positive values are guaranteed to tell you that you have an acute-angled view.
It is impossible not to say about the right triangle. This is the most ideal view, where all the intersection points of the medians, bisectors and heights coincide. The center of the inscribed and circumscribed circle also lies in one place. To solve problems, you only need to know one side, since the angles are initially given to you, and the other two sides are known. That is, the figure is specified by only one parameter. There are isosceles triangles. Their main feature is the equality of two sides and angles at the base.
Sometimes the question is whether there is a triangle with given sides. In fact, you are asked if this description fits the main types. 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 in the task they are asked to find the cosines of the angles of a triangle with sides 3,5,9, then here is an obvious catch. This can be explained without complicated mathematical techniques. Suppose you want to go from point A to point B. The straight line distance 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 moving through the store, you will pass one kilometer less. But since point C is not located on line AB, you will have to go the extra distance. A contradiction arises here. This, of course, is a conditional explanation. Mathematics knows more than one way of proving that all kinds of triangles obey the basic identity. It states that the sum of the two sides is greater than the length of the third.
Any species has the following properties:
1) The sum of all angles is 180 degrees.
2) There is always an orthocenter - the intersection point of all three heights.
3) All three medians drawn from the vertices of the inner corners intersect in one place.
4) A circle can be described around any triangle. You can also enter a circle so that it has only three points of contact and does not go beyond the outside.
Now you have become acquainted with the basic properties that various types of triangles possess. In the future, it is important to understand what you are dealing with when solving a problem.