Define an acute triangle. Types of triangles: rectangular, acute, obtuse
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internal combustion ...) Aqueous solutions for passivation. Aqueous solutions for etching - removing oxides from the surface Aqueous solutions for phosphating Aqueous solutions and mixtures for chemical oxidation and coloring of metals. Aqueous solutions and mixtures for chemical polishing Degreasing aqueous solutions and organic solvents pH value. pH tables., but small. So, for example, a triangle with vertices labeled A, B and C has sides a, b, c.
If we consider a triangle in Euclidean space, then it is a geometric figure that is formed using three segments connecting three points that do not lie on the same straight line.
Look carefully at the picture shown above. On it, points A, B and C are the vertices of this triangle, and its segments are called the sides of the triangle. Each vertex of this polygon forms angles inside it.
Types of triangles
According to the size of the angles of triangles, they are divided into such varieties as: Rectangular;
Acute angular;
Obtuse.
Rectangular triangles include those that have one right angle and the other two have acute angles.
Acute triangles are those in which all its angles are acute.
And if a triangle has one obtuse angle and the other two acute angles, then such a triangle is classified as obtuse.
Each of you understands perfectly well that not all triangles have equal sides. And according to the length of its sides, triangles can be divided into:
Isosceles;
Equilateral;
Versatile.
Task: Draw different types triangles. Define them. What difference do you see between them?
Basic properties of triangles
Although these simple polygons may differ from each other in the size of their angles or sides, each triangle has the basic properties that are characteristic of this figure.
In any triangle:
The total sum of all its angles is 180º.
If it belongs to equilaterals, then each of its angles is 60º.
An equilateral triangle has equal and equal angles.
The smaller the side of the polygon, the smaller the angle opposite it, and vice versa, the larger angle is opposite the larger side.
If the sides are equal, then opposite them are equal angles, and vice versa.
If we take a triangle and extend its side, we end up with an external angle. He equal to the sum internal corners.
In any triangle, its side, no matter which one you choose, will still be less than the sum of the other 2 sides, but more than their difference:
1. a< b + c, a >b–c;
2. b< a + c, b >a–c;
3.c< a + b, c >a–b.
Exercise
The table shows the already known two angles of the triangle. Knowing the total sum of all angles, find what the third angle of the triangle is equal to and enter it into the table:
1. How many degrees does the third angle have?
2. What type of triangle does it belong to?
Tests for equivalence of triangles
I sign
II sign
III sign
Height, bisector and median of a triangle
The altitude of a triangle - the perpendicular drawn from the vertex of the figure to its opposite side is called the altitude of the triangle. All altitudes of a triangle intersect at one point. The point of intersection of all 3 altitudes of a triangle is its orthocenter.
A segment drawn from a given vertex and connecting it at the middle of the opposite side is the median. Medians, as well as altitudes of a triangle, have one common point of intersection, the so-called center of gravity of the triangle or centroid.
The bisector of a triangle is a segment connecting the vertex of an angle and a point on the opposite side, and also dividing this angle in half. All bisectors of a triangle intersect at one point, which is called the center of the circle inscribed in the triangle.
The segment that connects the midpoints of 2 sides of a triangle is called the midline.
Historical reference
A figure such as a triangle was known back in Ancient times. This figure and its properties were mentioned on Egyptian papyri four thousand years ago. A little later, thanks to the Pythagorean theorem and Heron’s formula, the study of the properties of a triangle moved to more high level, but still, this happened more than two thousand years ago.
In the 15th – 16th centuries, a lot of research began to be carried out on the properties of a triangle, and as a result, a science such as planimetry arose, which was called “New Triangle Geometry”.
Russian scientist N.I. Lobachevsky made a huge contribution to the knowledge of the properties of triangles. His works later found application in mathematics, physics and cybernetics.
Thanks to knowledge of the properties of triangles, such a science as trigonometry arose. It turned out to be necessary for a person in his practical needs, since its use is simply necessary when drawing up maps, measuring areas, and even when designing various mechanisms.
What is the most famous triangle you know? This is of course the Bermuda Triangle! It got its name in the 50s because geographical location points (vertices of the triangle), within which, according to the existing theory, associated anomalies arose. The vertices of the Bermuda Triangle are Bermuda, Florida and Puerto Rico.
Assignment: What theories about the Bermuda Triangle have you heard?
Did you know that in Lobachevsky’s theory, when adding the angles of a triangle, their sum always has a result less than 180º. In Riemann's geometry, the sum of all the angles of a triangle is greater than 180º, and in the works of Euclid it is equal to 180 degrees.
Homework
Solve a crossword puzzle on a given topic
Questions for the crossword:
1. What is the name of the perpendicular that is drawn from the vertex of the triangle to the straight line located on the opposite side?
2. How, in one word, can you call the sum of the lengths of the sides of a triangle?
3. Name a triangle whose two sides are equal?
4. Name a triangle that has an angle equal to 90°?
5. What is the name of the largest side of the triangle?
6. What is the name of the side of an isosceles triangle?
7. There are always three of them in any triangle.
8. What is the name of a triangle in which one of the angles exceeds 90°?
9. The name of the segment connecting the top of our figure with the middle of the opposite side?
10. In a simple polygon ABC, the capital letter A is...?
11. What is the name of the segment dividing the angle of a triangle in half?
Questions on the topic of triangles:
1. Define it.
2. How many heights does it have?
3. How many bisectors does a triangle have?
4. What is its sum of angles?
5. What types of this simple polygon do you know?
6. Name the points of the triangles that are called remarkable.
7. What device can you use to measure the angle?
8. If the clock hands show 21 o'clock. What angle do the hour hands make?
9. At what angle does a person turn if he is given the command “left”, “circle”?
10. What other definitions do you know that are associated with a figure that has three angles and three sides?
A triangle (from the point of view of Euclidean space) is a geometric figure that is formed by three segments connecting three points that do not lie on the same straight line. The three points that formed the triangle are called its vertices, and the segments connecting the vertices are called the sides of the triangle. What types of triangles are there?
Equal triangles
There are three signs that triangles are equal. Which triangles are called equal? These are those who:
- two sides and the angle between these sides are equal;
- one side and two adjacent angles are equal;
- all three sides are equal.
Right triangles have following signs equality:
- By sharp corner and hypotenuse;
- along an acute angle and leg;
- on two legs;
- along the hypotenuse and leg.
What types of triangles are there?
By number equal sides a triangle can be:
- Equilateral. This is a triangle with three equal sides. All angles in an equilateral triangle are equal to 60 degrees. In addition, the centers of the circumscribed and inscribed circles coincide.
- Unequilateral. A triangle that has no equal sides.
- Isosceles. This is a triangle with two equal sides. Two identical sides are the sides, and the third side is the base. In such a triangle, the bisector, median and altitude coincide if they are lowered to the base.
According to the size of the angles, a triangle can be:
- Obtuse - when one of the angles is more than 90 degrees, that is, when it is obtuse.
- Acute - if all three angles in the triangle are acute, that is, they measure less than 90 degrees.
- Which triangle is called a right triangle? This is one that has one right angle equal to 90 degrees. The two sides that form this angle will be called the legs, and the hypotenuse will be the side opposite the right angle.
Basic properties of triangles
- The smaller angle always lies opposite the smaller side, and the larger angle always lies opposite the larger side.
- Equal angles always lie opposite equal sides, but opposite different sides There are always different angles. In particular, in an equilateral triangle, all angles have the same value.
- In any triangle, the sum of the angles is 180 degrees.
- An external angle can be obtained by extending one of the sides of a triangle. The size of the external angle will be equal to the sum of the internal angles not adjacent to it.
- A side of a triangle is greater than the difference of its two other sides, but less than their sum.
In Lobachevsky's spatial geometry, the sum of the angles of a triangle will always be less than 180 degrees. On a sphere this value is more than 180 degrees. The difference between 180 degrees and the sum of the angles of the triangle is called a defect.
Perhaps the most basic, simple and interesting figure in geometry is the triangle. I know high school its basic properties are studied, but sometimes knowledge on this topic is incomplete. The types of triangles initially determine their properties. But this view remains mixed. Therefore, now let’s look at this topic in a little more detail.
The types of triangles depend on degree measure corners These figures are acute, rectangular and obtuse. If all angles do not exceed 90 degrees, then the figure can safely be called acute. 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 one under consideration is called obtuse-angled.
There are many problems for acute-angled subtypes. Distinctive feature is the internal location of the intersection points of bisectors, medians and altitudes. In other cases, this condition may not be met. It is not difficult to determine the type of triangle figure. It is enough to know, for example, the cosine of each angle. If any values are less than zero, then the triangle is in any case obtuse. In the case of a zero indicator, the figure has a right angle. All positive values are guaranteed to tell you that you are looking at an angular view.
One cannot help but mention the regular 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 circle also lies in the same place. To solve problems, you need to know only one side, since the angles are initially given to you, and the other two sides are known. That is, the figure is specified by only one parameter. They exist main feature- equality of two sides and angles at the base.
Sometimes the question arises as to whether a triangle with given sides exists. In fact you are asked if it is suitable this description under the main types. 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 you to find the cosines of the angles of a triangle with sides of 3,5,9, then the obvious can be explained without complex mathematical techniques. 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 is 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 straight AB, you will have to walk an extra distance. There is a contradiction here. This is, of course, a conditional explanation. Mathematics knows more than one way to prove that all types of triangles obey the basic identity. It states that the sum of two sides longer third.
Any 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 drawn around any triangle. You can also inscribe a circle so that it has only three points of contact and does not extend beyond the outer sides.
Now you are familiar with the basic properties that different types of triangles have. In the future, it is important to understand what you are dealing with when solving a problem.
Today we are going to the country of Geometry, where we will get acquainted with various types triangles.
Consider geometric figures and find the “extra” one among them (Fig. 1).
Rice. 1. Illustration for example
We see that figures No. 1, 2, 3, 5 are quadrilaterals. Each of them has its own name (Fig. 2).
Rice. 2. Quadrilaterals
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 line and three segments connecting these points in pairs.
The points are called vertices of the triangle, 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. According to the size of the angle, triangles are 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 rectangular 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, that is, more than 90° (Fig. 6).
Rice. 6. Obtuse triangle
Based on the number of equal sides, triangles are equilateral, isosceles, scalene.
An isosceles triangle is one 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 base angles are equal.
There are isosceles triangles acute and obtuse(Fig. 8) .
Rice. 8. Acute and obtuse isosceles triangles
An equilateral triangle is one 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 scalene triangle is one in which all three sides have different lengths(Fig. 10).
Rice. 10. Scalene triangle
Complete the task. Distribute 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: No. 2, No. 6.
Obtuse triangles: No. 4, No. 5.
We will distribute the same triangles 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.
Look at the pictures.
Think about what piece of wire each triangle was made from (Fig. 12).
Rice. 12. Illustration for the task
You can think like this.
The first piece of wire is divided into three equal parts, so you can make an equilateral triangle from it. He is shown third in the picture.
The second piece of wire is divided into three different parts, so it can be used to make a scalene triangle. It is shown first in the picture.
The third piece of wire is divided into three parts, where two parts have the same length, which means that an isosceles triangle can be made from it. In the picture he is shown second.
Today in class we learned about different types of triangles.
Bibliography
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
- M.I. Moro. Math lessons: Guidelines for the teacher. 3rd grade. - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
- "School of Russia": Programs for primary school. - M.: “Enlightenment”, 2011.
- S.I. Volkova. Mathematics: Test work. 3rd grade. - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.
- Nsportal.ru ().
- Prosv.ru ().
- Do.gendocs.ru ().
Homework
1. Complete the phrases.
a) A triangle is a figure that consists of ... that do not lie on the same line, and ... that connect these points in pairs.
b) The points are called … , segments - his … . The sides of the triangle form at the vertices of the triangle ….
c) According to the size of the angle, triangles are ... , ... , ... .
d) Based on the number of equal sides, triangles are ... , ... , ... .
2. Draw
a) right triangle;
b) acute triangle;
c) obtuse triangle;
d) equilateral triangle;
e) scalene triangle;
e) isosceles triangle.
3. Create an assignment on the topic of the lesson for your friends.
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