Triangle equilateral: properties, attributes, area, perimeter

In the school course of geometry, a huge amount of time is devoted to the study of triangles. Students calculate angles, build bisectrixes and heights, find out what the figures are different from each other, and how the easiest way is to find their area and perimeter. It seems that this is not useful in life, but sometimes it is still useful to know, for example, how to determine that the triangle is equilateral or obtuse. How can this be done?

Types of triangles

Three points that do not lie on one line, and the segments that connect them. It seems that this figure is the simplest. What kind of triangles can there be if they have only three sides? In fact, the options are quite large, and some of them are given special attention in the school course of geometry. The right triangle is equilateral, that is, all its angles and sides are equal. He has a number of remarkable properties, which will be discussed further.

In an isosceles one, only two sides are equal, and it is also quite interesting. In a rectangular and obtuse triangle, as one can easily guess, respectively, one of the corners is straight or blunt. 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. In this case, it is rectangular. It is believed that such a triangle was actively used by Egyptian surveyors and architects to build right angles. There is an opinion that with his help the famous pyramids were erected.

And yet, all the vertices of a triangle can lie on one straight line. In this case, it will be called degenerate, while all the others are nondegenerate. They are one of the subjects of the study of geometry.

Triangle equilateral

Of course, the correct figures are always the most interesting. They seem more perfect, more elegant. The formulas for calculating their characteristics are often simpler and shorter than for ordinary figures. This also applies to triangles. Not surprisingly, when studying geometry, they are given a lot of attention: schoolchildren are taught to distinguish correct figures from the others, and also talk about some of their interesting characteristics.

Signs and properties

As it is not difficult to guess from the title, each side of an equilateral triangle is equal to the other two. In addition, it has a number of features, through which you can determine whether the correct figure or not.

• All its angles are equal, their magnitude is 60 degrees;
• Bisectors, heights and medians drawn from each vertex coincide;
• The right triangle has 3 axes of symmetry, it does not change when rotating by 120 degrees.
• The center of the inscribed circle is also the center of the circumscribed circle and the point of intersection of 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 assertions mentioned are valid.

All triangles have a number of remarkable properties. Firstly, the middle line, that is, the segment dividing the two sides in half and parallel to the third, is equal to half the base. Secondly, the sum of all the angles of this figure is always 180 degrees. In addition, there is another interesting interrelation in triangles. So, against the larger side lies a larger angle and vice versa. But this, of course, has no relation to an equilateral triangle, because all angles are equal.

Inscribed and circumscribed circles

Often in the course of geometry, students also learn how the figures can interact with each other. In particular, we study circles that are inscribed into polygons or described near them. What is it about?

Inscribed is a circle for which all sides of the polygon are tangent. Described - the one that has points of contact with all angles. For each triangle, it is always possible to construct both the first and second circles, but only one of each kind. The evidence of these two Theorems are given in the school course of geometry.

In addition to calculating the parameters of the triangles themselves, some problems also involve calculating the radii of these circles. And the formulas applied to
An equilateral triangle are as follows:

R = a / √ ̅3;

R = a / 2√ ̅3;

Where r is the radius of the inscribed circle, R is the radius of the circumscribed circle, and a is the length of the side of the triangle.

Calculating height, perimeter and area

The main parameters, calculated by students during the study of geometry, remain unchanged for almost any figure. This is the perimeter, area and height. For simplicity of calculations, there are various formulas.

So, the perimeter, that is, the length of all sides, is calculated in the following ways:

P = 3a = 3√ ̅3R = 6√ ̅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 = (√ ̅3 / 2) * a, where a is the side length.

Finally, the formula of the area of an equilateral triangle is derived from the standard, that is, the product of half the base on its height.

S = (√ ̅3 / 4) * a ² , where a is the side length.

Also, this value can be calculated through the parameters of the circumscribed or inscribed circle. There are also special formulas for this:

S = 3√ ̅3r ² = (3√ ̅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 shape using the minimal set
Tools: compasses and ruler without divisions.

In order to build the right triangle with just these tools, you need to perform several steps.

1. You need to draw a circle with any radius and centered at an arbitrary point A. It should be noted.
2. Next, you need to draw a straight line through this point.
3. The intersections of the circle and the line must be designated as B and C. All constructions must be carried out with the greatest possible accuracy.
4. Next, we need to construct another circle with the same radius and center at point C or an arc with the corresponding parameters. The intersection points will be designated as D and F.
5. The points B, F, D must be joined by segments. The equilateral triangle is constructed.

Solving such problems usually presents a problem for schoolchildren, but this skill can be useful in everyday life.