In the school geometry course, a huge amount of time is devoted to the study of triangles. Students calculate angles, build bisectors and heights, find out how shapes differ from each other, and the easiest way to find their area and perimeter. It seems that this is not useful in any way in life, but sometimes it is still useful to know, for example, how to determine that a triangle is equilateral or obtuse. How to do it?
Types of Triangles
Three points that do not lie on the same straight line, and the segments that connect them. It seems that this figure is the simplest. What can triangles look like if they only have three sides? In fact, there are a fairly large number of options, and some of them are given special attention as part of the school geometry course. An equilateral triangle is an equilateral one, that is, all its angles and sides are equal. It has a number of remarkable properties, which will be discussed later.
The isosceles has only two equal sides, and it is also quite interesting. In right-angled and obtuse-angled triangles, as you might guess, respectively, one of the angles is right or obtuse. Atthis they can also be isosceles.
There is also a special kind of triangle called Egyptian. Its sides are 3, 4 and 5 units. However, 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 the famous pyramids were built with its help.
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 called non-degenerate. They are one of the subjects of study of geometry.
Equilateral triangle
Of course, correct figures are always the most interesting. They seem more perfect, more graceful. The 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 when studying geometry: schoolchildren are taught to distinguish regular figures from the rest, and also talk about some of their interesting characteristics.
Signs and properties
As you might guess from the name, each side of an equilateral triangle is equal to the other two. In addition, it has a number of features, thanks to which you can determine whether the figure is correct or not.
- all its angles are equal, their value is 60 degrees;
- bisectors, heights and medians drawn from each vertex are the same;
- regular triangle has 3 axes of symmetry, itdoes not change when rotated 120 degrees.
- the center of the inscribed circle is also the center of the circumscribed circle and the point of intersection of the medians, bisectors, heights and perpendicular bisectors.
If at least one of the above signs is observed, then the triangle is equilateral. For a regular figure, all the above statements are true.
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 equal to 180 degrees. In addition, there is another interesting relationship in triangles. So, opposite 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.
Inscribed and circumscribed circles
It is not uncommon for students in a geometry course to also learn how shapes can interact with each other. In particular, circles inscribed in polygons or described around them are studied. What is it about?
An inscribed circle is a circle for which all sides of the polygon are tangent. Described - the one that has points of contact with all corners. For each triangle, it is always possible to construct both the first and second circles, but only one of each type. Evidence for these two
theorems are given inschool 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 for the equilateral triangle look like this:
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.
Calculating height, perimeter and area
The main parameters, which are calculated by schoolchildren while studying geometry, remain unchanged for almost any figure. These are 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=3√ ̅3R=6√ ̅3r, where a is the side of a regular triangle, R is the radius of the circumcircle, r is the inscribed circle.
Height:
h=(√ ̅3/2)a, where a is the length of the side.
Finally, the formula for the area of an equilateral triangle is derived from the standard formula, that is, the product of half the base and its height.
S=(√ ̅3/4)a2, where a is the length of the side.
Also, this value can be calculated through the parameters of the circumscribed or inscribed circle. There are also special formulas for this:
S=3√ ̅3r2=(3√ ̅3/4)R2, where r and R are respectively the radii inscribed and circumscribed circles.
Building
One moreAn interesting type of task, including triangles, is associated with the need to draw one or another figure using the minimum set
tools: a compass and a ruler without divisions.
It takes a few steps to build a proper triangle with just these tools.
- You need to draw a circle with any radius and centered at an arbitrary point A. It must be marked.
- Next, you need to draw a straight line through this point.
- Intersections of a circle and a straight line must be designated as B and C. All constructions must be carried out with the greatest possible accuracy.
- Next, you need to build another circle with the same radius and center at point C or an arc with the appropriate parameters. Intersections will be marked as D and F.
- Points B, F, D must be connected by segments. An equilateral triangle is constructed.
Solving such problems is usually a problem for schoolchildren, but this skill can be useful in everyday life.
