In the school course of geometry, a huge amount of time is devoted to the study of triangles. Pupils calculate angles, build bisectors and heights, find out how the figures differ from each other, and how it is easiest to find their area and perimeter. It seems that this is not useful in life, but sometimes itβs still useful to find out, for example, how to determine whether a triangle is equilateral or obtuse. How to do this?
Types of Triangles
Three points that do not lie on one straight line, and segments that connect them. This figure seems to be the simplest. What can be triangles if they have only three sides? In fact, there are quite a lot of options, and some of them are given special attention as part of the school geometry course. A regular triangle is equilateral, that is, all its angles and sides are equal. It has a number of remarkable properties, which will be discussed later.
The isosceles have only two sides, and it is also quite interesting. In right- angled and obtuse-angled triangles, as one can easily guess, respectively, one of the angles is straight or obtuse. However, they can also be isosceles.
There is also a special kind of triangle, called Egyptian. Its sides are equal to 3, 4 and 5 units. Moreover, it is rectangular. It is believed that such a triangle was actively used by Egyptian surveyors and architects to build right angles. It is believed that with its help the famous pyramids were erected.
Nevertheless, all the vertices of a triangle can lie on one straight line. In this case, it will be called degenerate, while all the rest - non-degenerate. They are one of the subjects of the study of geometry.
Equilateral triangle
Of course, the correct figures are always of the greatest interest. They seem more perfect, more elegant. Formulas for calculating their characteristics are often simpler and shorter than for ordinary figures. This also applies to triangles. It is not surprising that a lot of attention is paid to them in the study of geometry: schoolchildren are taught to distinguish the correct figures from the rest, and also talk about some of their interesting characteristics.
Signs and properties
As the name suggests, each side of an equilateral triangle is equal to the other two. In addition, he has a number of signs, thanks to which you can determine whether the figure is correct or not.
- all its angles are equal, their size is 60 degrees;
- bisectors, heights and medians drawn from each vertex coincide;
- a regular triangle has 3 axes of symmetry; it does not change when rotated by 120 degrees.
- the center of the inscribed circle is also the center of the circumscribed circle and the intersection point of the medians, bisectors, heights and median perpendiculars.
If at least one of the above signs is observed, then the triangle is equilateral. For the correct figure, all the above statements are true.
All triangles have a number of remarkable properties. Firstly, the middle line, that is, a segment that divides two sides in half and parallel to the third, is equal to half the base. Secondly, the sum of all the corners of this figure is always equal to 180 degrees. In addition, another interesting relationship is observed in triangles. So, against the larger side lies a larger angle and vice versa. But this, of course, has nothing to do with an equilateral triangle, because it has all the angles equal.
Inscribed and circled circles
Often in a geometry course, students also learn how shapes can interact with each other. In particular, circles inscribed in polygons or circumscribed around them are studied. What is it about?
An inscribed circle is a circle for which all sides of the polygon are tangent. Described - one that has points of contact with all angles. For each triangle, you can always build both the first and second circles, but only one of each kind. Evidence of these two
theorems are given in the school geometry course.
In addition to calculating the parameters of the triangles themselves, some tasks also involve calculating the radii of these circles. And the formulas applied to
an equilateral triangle are as follows:
r = a / β Μ
3;
R = a / 2β Μ
3;
where r is the radius of the inscribed circle, R is the radius of the circumscribed circle, a is the length of the side of the triangle.
Calculation of height, perimeter and area
The main parameters that students are involved in calculating while studying geometry remain unchanged for almost any shape. This is the perimeter, area and height. For ease of calculation, there are various formulas.
So, the perimeter, that is, the length of all sides, is calculated in the following ways:
P = 3a = 3β Μ
3R = 6β Μ
3r, where a is the side of the regular triangle, R is the radius of the circumscribed circle, r is the inscribed one.
Height:
h = (β Μ
3 / 2) * a, where a is the length of the side.
Finally, the formula for the area of ββan equilateral triangle is derived from the standard, that is, the product of half the base and its height.
S = (β Μ
3 / 4) * a 2 , where a is the length of the side.
Also, this value can be calculated through the parameters of the circumscribed or inscribed circles. There are also special formulas for this:
S = 3β Μ
3r 2 = (3β Μ
3 / 4) * R 2 , where r and R are the radii of the inscribed and circumscribed circles, respectively.
Building
Another interesting type of problem, including triangles, is related to the need to draw a particular figure using a minimal set
instruments: compass and ruler without divisions.
In order to build a regular triangle using only these devices, you need to perform several steps.
- It is necessary to draw a circle with any radius and with the center at an arbitrary point A. It must be noted.
- Next you need to draw a straight line through this point.
- The intersection of a circle and a line must be designated as B and C. All constructions should be carried out with the greatest possible accuracy.
- Next, we need to build another circle with the same radius and center at point C or an arc with the corresponding parameters. The intersection will be indicated by D and F.
- Points B, F, D must be connected by segments. An equilateral triangle is built.
Solving such problems usually poses a problem for students, but this skill can be useful in everyday life.