in school geometry course a huge amount of time is devoted to the study of triangles.Students calculate the angles, bisector build and height, find out what the figures are different from each other, and how the easiest way to find their area and perimeter.It seems that it is not useful in life, but sometimes still useful to know, for example, determine that an equilateral triangle or obtuse.How to do it?
types of triangles
three points that do not lie on one line, and the segments that connect them.It seems that the figure - the most simple.What could be the triangles, if they have all three parties?In fact, quite a number of options, and some of them are given special attention in the school geometry course.Right triangle - equilateral, ie all its angles and sides are equal.He has a number of remarkable properties, which will be discussed further.
have isosceles are only two sides, and it is also quite interesting.In rectangular and an obtuse-angled triangles, as easy to guess, respectively, one of the angles is right or obtuse.However, they may also be isosceles.
There is a special kind of triangle, called the Egyptian.Its sides are 3, 4 and 5 units.He is rectangular.It is believed that a triangle was used extensively by the Egyptian land surveyors and architects to construct right angles.It is believed that with the help of the famous pyramids were built.
Still, all the vertices of a triangle can lie on a straight line.In this case, it will be called degenerate, while the rest - non-degenerate.That they are one of the subjects of the study of geometry.
equilateral triangle
course, correct figure always cause the greatest interest.They seem to be more sophisticated, more elegant.Formula calculation of their characteristics is often easier and shorter than for conventional shapes.This applies to triangles.Not surprisingly, the study of geometry, they paid a lot of attention: students are taught to distinguish the correct figure from the other, and talk about some of their interesting characteristics.
characteristics and properties
As you might guess from the title, each side of the equilateral triangle is equal to the other two.In addition, it has a number of features by which it can be determined whether the correct figure or not.
- all its angles are equal, their value is 60 degrees;
- bisector, height and median drawn from each vertex are the same;
- equilateral triangle has three axes of symmetry, it does not change when you turn 120 degrees.
- center of the inscribed circle is also the center of the circumscribed circle and the point of intersection of the medians, bisectors, heights and midperpendicular.
If there is at least one of the above features, the triangle - equilateral.For the correct figure all these allegations are true.
All the triangles have a number of remarkable properties.Firstly, the middle line, then a segment dividing in half and two sides parallel to the third, is equal to half the base.Secondly, the sum of all the angles of this shape is always equal to 180 degrees.In addition, the triangle is observed another curious relationship.Thus, against the larger side is greater angle and vice versa.But this, of course, to an equilateral triangle is not relevant, because it has all the angles are equal.
inscribed and circumscribed circles
Often in the course of geometry, students also learn how the pieces can interact with each other.In particular, the study of the circle inscribed in polygons or disclosed about them.What is it about?
inscribed call this circle, for which all sides of the polygon are tangents.It describes - one that has points of contact with all the angles.For each triangle is always possible to construct both the first and the second circle, but only one of each kind.The proofs of these two theorems are given in the school geometry course.
addition to calculating the parameters themselves triangles, some problems also involve the calculation of the radii of the circles.And the formula applied to
equilateral triangle as follows:
r = a / √ ̅3;
R = a / 2√ ̅3;
where r - radius of the inscribed circle, R - the radius of the circle, a - the length of the sides of the triangle.
Calculating the height of the perimeter and area
main parameters involved in the calculation of which the students while learning geometry remain unchanged for virtually any figure.This perimeter, area and height.To simplify the calculations there are various formulas.
So, the perimeter, it is the length of all sides is calculated in the following ways:
P = 3a = 3√ ̅3R = 6√ ̅3r, where a - side of the equilateral triangle, R - the radius of the circle, r - inscribed.
Height:
h = (√ ̅3 / 2) * a, where a - length of the side.
Finally, the formula for the area of an equilateral triangle is derived from the standard, that is, half of the work on the grounds of his height.
S = (√ ̅3 / 4) * a2, where a - side length.
Also this value can be calculated through the parameters described or inscribed circle.To do this, there are also special formulas:
S = 3√ ̅3r2 = (3√ ̅3 / 4) * R2, where r and R - the radii of the inscribed and circumscribed circles.
Building
Another interesting type of tasks concerning including triangles, linked to the need to draw this or that figure, using a minimal set of
tools: a compass and a ruler without divisions.
In order to construct an equilateral triangle with just these devices, you must follow a few steps.
- necessary to draw a circle with any radius and centered at an arbitrarily chosen point A. It must be noted.
- Next you need to draw a line through this point.
- intersection of the circle and the line must be designated as B and C. All constructions must be conducted with the greatest possible precision.
- Next you need to build another circle with the same radius and center point C or arc with the appropriate parameters.Designated intersections will be designated as D and F.
- points B, F, D must be connected segments.An equilateral triangle is constructed.
solution of such problems is usually a problem for students, but this skill can be useful in everyday life.
