In the school course of geometry, a large numbertime is devoted to the study of triangles. Students calculate angles, build bisectrixes and heights, find out what the figures differ from each other, and how easy it 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?
Three points that do not lie on one line, andthe 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, there are quite a number of options, 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 noteworthy properties, which will be discussed further.
In an isosceles one, only two sides are equal, andalso quite interesting. In rectangular and obtuse-angled triangles, as it is easy to guess, respectively, one of the corners is straight or obtuse. However, they can also be isosceles.
There is also a special type of triangle calledEgyptian. Its sides are 3, 4, and 5 units. At the same time it is rectangular. It is believed that such a triangle was actively used by Egyptian land surveyors and architects to build right angles. It is believed that with its help the famous pyramids were erected.
Still, all the vertices of the triangle may lieon one straight line. In this case, it will be called degenerate, while all others will be called nondegenerate. They are one of the subjects of the study of geometry.
Of course, the correct figures always causegreatest interest. They seem more perfect, more graceful. Formulas for calculating their characteristics are often simpler and shorter than for ordinary shapes. This also applies to triangles. It is not surprising that in the study of geometry they are given quite a lot of attention: students are taught to distinguish the correct figures from the rest, and also tell about some of their interesting characteristics.
As you might guess from the name, eachthe side of an equilateral triangle is equal to two others. In addition, it has a number of features that make it possible to determine whether the figure is correct or not.
If there is at least one of the above signs, then the triangle is equilateral. For the correct figure, all the statements mentioned are true.
All triangles possess a number of noteworthyproperties. First, the middle line, that is, the segment dividing the two sides in half and parallel to the third, is half the base. Secondly, the sum of all angles of this figure is always 180 degrees. In addition, there is another interesting relationship in the 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 all its angles are equal.
Often in the geometry course, students also learnhow the shapes can interact with each other. In particular, the circles inscribed in polygons or described around them are studied. What is this about?
An inscribed circle is one for whichall sides of the polygon are tangent. Described - the one that has points of contact with all angles. For each triangle you can always construct both the first and the second circle, but only one of each kind. The evidence of these two
In addition to calculating the parameters of the triangles themselves, some tasks also involve the calculation of the radii of these circles. And the formulas applied to
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, a is the length of the side of the triangle.
The main parameters, the calculation of whichpupils are engaged in the study of geometry, remain unchanged for almost any shape. This is the perimeter, area and height. For ease of calculation, there are various formulas.
P = 3a = 3√ ̅3R = 6√ ̅3r, where a is the side of a 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 to its height.
S = (√ ̅3 / 4) * a2where a is the side length.
Also, this value can be calculated through the parameters of a circumscribed or inscribed circle. There are also special formulas for this:
S = 3√ r3r2 = (3√ ̅3 / 4) * R2, where r and R are respectively the radii of the inscribed and circumscribed circles.
Another interesting type of tasks, including triangles, is associated with the need to draw a particular shape using the minimum set
In order to build a regular triangle using only these devices, several steps are necessary.
Solving such problems is usually a problem for schoolchildren, but this skill can be useful in everyday life.
