When studying mathematics, students begin to get acquainted with various types of geometric shapes. Today we will talk about different types of triangles.
Definition
Geometric figures that consist of three points that are not on the same straight line are called triangles.
The line segments connecting the points are called sides, and the points are called vertices. Vertices are marked with big with Latin letters, for example: A, B, C.
The sides are indicated by the names of the two points of which they consist - AB, BC, AC. Intersecting, the sides form angles. The bottom side is considered the base of the figure.
Rice. 1. Triangle ABC.
Types of triangles
Triangles are classified according to angles and sides. Each type of triangle has its own properties.
There are three types of triangles in the corners:
- acute-angled;
- rectangular;
- obtuse.
All angles acute-angled triangles are acute, that is, the degree measure of each is no more than 90 0.
Rectangular the triangle contains a right angle. The other two angles will always be acute, because otherwise the sum of the angles of the triangle will exceed 180 degrees, which is impossible. The side that is opposite the right angle is called the hypotenuse, and the other two legs. The hypotenuse is always greater than the leg.
obtuse the triangle contains an obtuse angle. That is, an angle greater than 90 degrees. The other two angles in such a triangle will be acute.
Rice. 2. Types of triangles in the corners.
A Pythagorean triangle is a rectangle whose sides are 3, 4, 5.
Moreover, the larger side is the hypotenuse.
Such triangles are often used to compose simple problems in geometry. Therefore, remember: if two sides of a triangle are 3, then the third one will definitely be 5. This will simplify the calculations.
Types of triangles on the sides:
- equilateral;
- isosceles;
- versatile.
Equilateral a triangle is a triangle in which all sides are equal. All angles of such a triangle are equal to 60 0, that is, it is always acute-angled.
Isosceles a triangle is a triangle with only two equal sides. These sides are called lateral, and the third - the base. In addition, the angles at the base of an isosceles triangle are equal and always acute.
Versatile or an arbitrary triangle is a triangle in which all lengths and all angles are not equal to each other.
If there are no clarifications about the figure in the problem, then it is generally accepted that we are talking about an arbitrary triangle.
Rice. 3. Types of triangles on the sides.
The sum of all the angles of a triangle, regardless of its type, is 1800.
Opposite the larger angle is the larger side. And also the length of any side is always less than the sum of its other two sides. These properties are confirmed by the triangle inequality theorem.
There is a concept of a golden triangle. it isosceles triangle, whose two sides are proportional to the base and equal to a certain number. In such a figure, the angles are proportional to the ratio 2:2:1.
A task:
Is there a triangle whose sides are 6 cm, 3 cm, 4 cm?
Solution:
To solve this task, you need to use the inequality a
What have we learned?
From this material from the 5th grade mathematics course, we learned that triangles are classified by sides and angles. Triangles have certain properties that can be used when solving problems.
Triangle is a polygon with 3 sides (or 3 corners). The sides of a triangle are often indicated by small letters, which correspond to capital letters denoting the reverse vertices.
Acute triangle A triangle is called if all three angles are acute.
obtuse triangle A triangle is called in which one of the angles is obtuse.
right triangle a triangle is called, in which one of the angles is a right one, in other words it is equal to 90 °; sides a, b forming a right angle are called legs; side c, the opposite of the right angle, is called hypotenuse.
Isosceles triangle a triangle is called, in which two of its sides are equal (a \u003d c); these equal sides are called lateral, the 3rd side is called the base of the triangle.
equilateral triangle a triangle is called, in which all its sides are equal (a \u003d b \u003d c). In that case, none of its sides (abc) is equal in a triangle, then this is unequal triangle.
The main characteristics of triangles
In any triangle:
Signs of equality of triangles
Triangles are congruent, in which case they are respectively equal:
Signs of equality of right triangles
Two right triangles are equal, in which case one of the following criteria is produced:
Heighttriangle is a perpendicular dropped from any vertex on reverse side(or its continuation). This side is called the base of the triangle. The three heights of a triangle always intersect at one point, called triangle orthocenter.
The orthocenter of an acute triangle is placed inside the triangle, and the orthocenter of an obtuse triangle is placed outside; The orthocenter of a right triangle coincides with the apex of the right angle.
Median is a line segment connecting any apex of a triangle with the midpoint of the reverse side. Three medians of a triangle intersect at one point, which always lies inside the triangle and is its center of mass. This point divides each median 2:1 from the top.
Bisector- this is a segment of the bisector of the angle from the vertex to the point of intersection with the back side. Three bisectors of a triangle intersect at one point, which always lies inside the triangle and is the center of the inscribed circle. The bisector divides the reverse side into parts proportional to the adjacent sides.
Median perpendicular is a perpendicular drawn from the midpoint of the segment (side). The three median perpendiculars of a triangle intersect at one point, which is the center of the circumscribed circle.
AT acute triangle this point lies inside the triangle, in an obtuse triangle - outside, in a rectangular one - in the middle of the hypotenuse. The orthocenter, center of mass, center of the circumscribed circle and the center of the inscribed circle coincide exclusively in an equilateral triangle.
Axiom of Pythagoras
AT right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
Confirmation of the Pythagorean axiom
Construct the square AKMB using the hypotenuse AB as the side. Then we continue the sides of the right triangle ABC so as to get the square CDEF, whose side is a + b. It is now clear that the area of square CDEF is (a + b) 2. On the other hand, this area is equal to the sum of the areas of four right triangles and square AKMB, in other words,
c 2 + 4 (ab / 2) = c 2 + 2 ab,
c 2 + 2 ab = (a + b) 2,
and we have:
c 2 = a 2 + b 2 .
Aspect ratio in a random triangle
AT general case(for a random triangle) we have:
c 2 \u003d a 2 + b 2 - 2 ab * cos C,
where C is the angle between sides a and b.
Additional to the site:
Today we are going to the country of Geometry, where we will get acquainted with different types of triangles.
Consider geometric figures and find among them the “extra” (Fig. 1).
Rice. 1. Illustration for example
We see that figures No. 1, 2, 3, 5 are quadrangles. Each of them has its own name (Fig. 2).
Rice. 2. Quadrangles
This means that the "extra" figure is a triangle (Fig. 3).
Rice. 3. Illustration for example
A triangle is a figure that consists of three points that do not lie on the same straight line, and three segments connecting these points in pairs.
The points are called triangle vertices, segments - his parties. The sides of the triangle form There are three angles at the vertices of a triangle.
The main features of a triangle are three sides and three corners. Triangles are classified according to the angle acute, rectangular and obtuse.
A triangle is called acute-angled if all three of its angles are acute, that is, less than 90 ° (Fig. 4).
Rice. 4. Acute triangle
A triangle is called right-angled if one of its angles is 90° (Fig. 5).
Rice. 5. Right Triangle
A triangle is called obtuse if one of its angles is obtuse, i.e. greater than 90° (Fig. 6).
Rice. 6. Obtuse Triangle
According to the number of equal sides, triangles are equilateral, isosceles, scalene.
An isosceles triangle is a triangle in which two sides are equal (Fig. 7).
Rice. 7. Isosceles triangle
These sides are called lateral, Third side - basis. In an isosceles triangle, the angles at the base are equal.
Isosceles triangles are acute and obtuse(Fig. 8) .
Rice. 8. Acute and obtuse isosceles triangles
An equilateral triangle is called, in which all three sides are equal (Fig. 9).
Rice. 9. Equilateral triangle
In an equilateral triangle all angles are equal. Equilateral triangles always acute-angled.
A triangle is called versatile, in which all three sides have different lengths (Fig. 10).
Rice. 10. Scalene triangle
Complete the task. Divide these triangles into three groups (Fig. 11).
Rice. 11. Illustration for the task
First, let's distribute according to the size of the angles.
Acute triangles: No. 1, No. 3.
Right triangles: #2, #6.
Obtuse triangles: #4, #5.
These triangles are divided into groups according to the number of equal sides.
Scalene triangles: No. 4, No. 6.
Isosceles triangles: No. 2, No. 3, No. 5.
Equilateral Triangle: No. 1.
Review the drawings.
Think about what piece of wire each triangle is made of (fig. 12).
Rice. 12. Illustration for the task
You can argue like this.
The first piece of wire is divided into three equal parts, so you can make an equilateral triangle out of it. It is shown third in the figure.
The second piece of wire is divided into three different parts, so you can make a scalene triangle out of it. It is shown first in the picture.
The third piece of wire is divided into three parts, where the two parts are the same length, so you can make an isosceles triangle out of it. It is shown second in the figure.
Today in the lesson we got acquainted with different types of triangles.
Bibliography
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
- M.I. Moreau. Math Lessons: Guidelines for the teacher. Grade 3 - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
- "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
- S.I. Volkov. Maths: Verification work. Grade 3 - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.
- Nsportal.ru ().
- Prosv.ru ().
- Do.gendocs.ru ().
Homework
1. Finish the phrases.
a) A triangle is a figure that consists of ..., not lying on the same straight line, and ..., connecting these points in pairs.
b) The points are called … , segments - his … . The sides of a triangle form at the vertices of a triangle ….
c) According to the size of the angle, triangles are ..., ..., ....
d) According to the number of equal sides, triangles are ..., ..., ....
2. Draw
a) a right triangle
b) an acute triangle;
c) an obtuse triangle;
d) an equilateral triangle;
e) scalene triangle;
e) an isosceles triangle.
3. Make a task on the topic of the lesson for your comrades.
Perhaps the most basic, simple and interesting figure in geometry is a triangle. I know high school its main properties are studied, but sometimes knowledge on this topic is formed incomplete. The types of triangles initially determine their properties. But this view remains mixed. So now let's take a closer look at this topic.
The types of triangles depend on the degree measure of the angles. These figures are acute, rectangular and obtuse. If all angles do not exceed 90 degrees, then the figure can safely be called acute-angled. If at least one angle of the triangle is 90 degrees, then you are dealing with a rectangular subspecies. Accordingly, in all other cases, the considered one is called obtuse-angled.
There are many tasks for acute-angled subspecies. hallmark is the interior location of the intersection points of the bisectors, medians, and heights. In other cases, this condition may not be met. Determining the type of figure "triangle" is not difficult. It is enough to know, for example, the cosine of each angle. If any values less than zero, so the triangle is obtuse in any case. In the case of a zero exponent, the figure has a right angle. All positive values are guaranteed to tell you that you have an acute-angled view.
It cannot be said about right triangle. This is the most ideal view, where all the intersection points of medians, bisectors and heights coincide. The center of the inscribed and circumscribed circles also lies in the same place. To solve problems, you need to know only one side, since the angles are initially set for you, and the other two sides are known. That is, the figure is given by only one parameter. There Are Them main feature- equality of two sides and angles at the base.
Sometimes there is a question about whether there is a triangle with given sides. What you are really asking is whether this description fits the main species. For example, if the sum of two sides is less than the third, then in reality such a figure does not exist at all. If the task asks to find the cosines of the angles of a triangle with sides 3,5,9, then the obvious can be explained here without complex mathematical tricks. Suppose you want to get from point A to point B. The distance in a straight line is 9 kilometers. However, you remembered that you need to go to point C in the store. The distance from A to C is 3 kilometers, and from C to B - 5. Thus, it turns out that when moving through the store, you will walk one kilometer less. But since point C is not located on line AB, you will have to go an extra distance. Here a contradiction arises. This is, of course, a hypothetical explanation. Mathematics knows more than one way to prove that all kinds of triangles obey the basic identity. It says that the sum of two sides is greater than the length of the third.
Each type has the following properties:
1) The sum of all angles is 180 degrees.
2) There is always an orthocenter - the point of intersection of all three heights.
3) All three medians drawn from the vertices of the interior angles intersect in one place.
4) A circle can be circumscribed around any triangle. It is also possible to inscribe a circle so that it has only three points of contact and does not go beyond the outer sides.
Now you are familiar with the main properties that different kinds triangles. In the future, it is important to understand what you are dealing with when solving a problem.
triangles
triangle A figure is called a figure that consists of three points that do not lie on one straight line, and three segments connecting these points in pairs. The points are called peaks triangle, and the segments - its parties.
Types of triangles
The triangle is called isosceles if its two sides are equal. These equal sides are called sides, and the third party is called basis triangle.
A triangle in which all sides are equal is called equilateral or correct.
The triangle is called rectangular, if it has a right angle, then there is a 90° angle. The side of a right triangle opposite the right angle is called hypotenuse the other two sides are called legs.
The triangle is called acute-angled if all three of its angles are acute, that is, less than 90 °.
The triangle is called obtuse, if one of its angles is obtuse, i.e. greater than 90°.
The main lines of the triangle
Median
Median triangle is a line segment that connects the vertex of a triangle with the midpoint of the opposite side of this triangle. 
Triangle median properties
The median divides the triangle into two triangles of the same area.
The medians of a triangle intersect at one point, which divides each of them in a ratio of 2:1, counting from the top. This point is called center of gravity triangle.
The entire triangle is divided by its medians into six equal triangles.
Bisector
Angle bisector is a ray that comes from its vertex, passes between its sides and bisects the given angle. Triangle bisector A segment of the bisector of an angle of a triangle connecting a vertex to a point on the opposite side of the triangle is called. 
Triangle bisector properties
Height
Height triangle is called a perpendicular drawn from the vertex of the triangle to the line containing the opposite side of this triangle. 
Triangle height properties
AT right triangle the height drawn from the vertex of a right angle divides it into two triangles, similar original.
AT acute triangle its two heights cut off from it similar triangles.
Median perpendicular
A line passing through the midpoint of a segment perpendicular to it is called perpendicular bisector to the segment .
Properties of the perpendicular bisectors of a triangle
Each point of the perpendicular bisector to a segment is equidistant from the ends of this segment. The converse statement is also true: each point equidistant from the ends of the segment lies on the perpendicular bisector to it.
The point of intersection of the midperpendiculars drawn to the sides of the triangle is the center circle circumscribed about this triangle.
middle line
The middle line of the triangle A line segment joining the midpoints of two of its sides is called.
Property of the midline of a triangle
The midline of a triangle is parallel to one of its sides and equal to half of that side.
Formulas and ratios
Signs of equality of triangles
Two triangles are congruent if they are respectively congruent:
two sides and the angle between them;
two corners and a side adjacent to them;
three sides.
Signs of equality of right triangles
Two right triangle are equal if they are respectively equal: 
hypotenuse and acute angle
leg and the opposite corner;
leg and adjacent angle;
two leg;
hypotenuse and leg.
similarity of triangles
Two triangles are similar if one of the following conditions is met, called signs of similarity:
two angles of one triangle are equal to two angles of another triangle;
two sides of one triangle are proportional to two sides of another triangle, and the angles formed by these sides are equal;
the three sides of one triangle are respectively proportional to the three sides of the other triangle.
In similar triangles, the corresponding lines ( heights, medians, bisectors etc.) are proportional.
Sine theorem
The sides of a triangle are proportional to the sines of the opposite angles, and the coefficient of proportionality is diameter
circle circumscribed about a triangle:
Cosine theorem
The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides times the cosine of the angle between them:
a 2 = b 2 + c 2 - 2bc cos
Triangle area formulas
Arbitrary triangle
a, b, c - sides; - angle between sides a and b; - semi-perimeter; R- radius of the circumscribed circle; r- radius of the inscribed circle; S- square; h a - height to side a.
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