The science of geometry tells us what a triangle, square, and cube are. In the modern world, everyone without exception studies it in schools. Also, the science that studies directly what a triangle is and what properties it has is trigonometry. She explores in detail all phenomena related to data. We will talk about what a triangle is today in our article. Their types will be described below, as well as some theorems associated with them.
What is a triangle? Definition
This is a flat polygon. It has three corners, as is clear from its name. It also has three sides and three vertices, the first of them are segments, the second are points. Knowing what two angles are equal to, you can find the third by subtracting the sum of the first two from the number 180.
What types of triangles are there?
They can be classified according to various criteria.
First of all, they are divided into acute-angled, obtuse-angled and rectangular. The first ones have sharp corners, that is, those that are equal to less than 90 degrees. In obtuse angles, one of the angles is obtuse, that is, one that is equal to more than 90 degrees, the other two are acute. TO acute triangles equilateral ones also apply. Such triangles have all sides and angles equal. They are all equal to 60 degrees, this can be easily calculated by dividing the sum of all angles (180) by three.
Right triangle
It is impossible not to talk about what a right triangle is.
Such a figure has one angle equal to 90 degrees (straight), that is, two of its sides are perpendicular. The remaining two angles are acute. They can be equal, then it will be isosceles. The Pythagorean theorem is related to the right triangle. Using it, you can find the third side, knowing the first two. According to this theorem, if you add the square of one leg to the square of the other, you can get the square of the hypotenuse. The square of the leg can be calculated by subtracting the square of the known leg from the square of the hypotenuse. Speaking about what a triangle is, we can also recall an isosceles triangle. This is one in which two of the sides are equal, and two angles are also equal.
What are leg and hypotenuse?
A leg is one of the sides of a triangle that forms an angle of 90 degrees. The hypotenuse is the remaining side that is opposite right angle. You can lower a perpendicular from it onto the leg. The ratio of the adjacent side to the hypotenuse is called cosine, and the opposite side is called sine.
- what are its features?
It's rectangular. Its legs are three and four, and its hypotenuse is five. If you see that the legs of a given triangle are equal to three and four, you can rest assured that the hypotenuse will be equal to five. Also, using this principle, you can easily determine that the leg will be equal to three if the second is equal to four, and the hypotenuse is equal to five. To prove this statement, you can apply the Pythagorean theorem. If two legs are equal to 3 and 4, then 9 + 16 = 25, the root of 25 is 5, that is, the hypotenuse is equal to 5. An Egyptian triangle is also a right triangle whose sides are 6, 8 and 10; 9, 12 and 15 and other numbers with the ratio 3:4:5.
What else could a triangle be?
Triangles can also be inscribed or circumscribed. The figure around which the circle is described is called inscribed; all its vertices are points lying on the circle. A circumscribed triangle is one into which a circle is inscribed. All its sides come into contact with it at certain points.
How is it located?
The area of any figure is measured in square units(sq. meters, sq. millimeters, sq. centimeters, sq. decimeters, etc.) This value can be calculated in a variety of ways, depending on the type of triangle. The area of any figure with angles can be found by multiplying its side by the perpendicular dropped onto it from opposite corner, and dividing this number by two. You can also find this value by multiplying the two sides. Then multiply this number by the sine of the angle located between these sides, and divide this result by two. Knowing all the sides of a triangle, but not knowing its angles, you can find the area in another way. To do this you need to find half the perimeter. Then alternately subtract from this number different sides and multiply the resulting four values. Next, find from the number that came out. The area of an inscribed triangle can be found by multiplying all the sides and dividing the resulting number by that circumscribed around it, multiplied by four.
The area of a circumscribed triangle is found in this way: we multiply half the perimeter by the radius of the circle that is inscribed in it. If then its area can be found as follows: square the side, multiply the resulting figure by the root of three, then divide this number by four. In a similar way, you can calculate the height of a triangle in which all sides are equal; to do this, you need to multiply one of them by the root of three, and then divide this number by two.
Theorems related to triangle
The main theorems that are associated with this figure are the Pythagorean theorem described above and cosines. The second (of sines) is that if you divide any side by the sine of the angle opposite it, you can get the radius of the circle that is described around it, multiplied by two. The third (cosines) is that if from the sum of the squares of the two sides we subtract their product multiplied by two and the cosine of the angle located between them, then we get the square of the third side.
Dali Triangle - what is it?
Many, when faced with this concept, at first think that this is some kind of definition in geometry, but this is not at all the case. Dali's triangle is common name three places that are closely connected with the life of the famous artist. Its “peaks” are the house in which Salvador Dali lived, the castle that he gave to his wife, as well as the museum of surrealist paintings. You can learn a lot during a tour of these places. interesting facts about this unique creative artist known throughout the world.
Triangle - definition and general concepts
A triangle is a simple polygon consisting of three sides and having the same number of angles. Its planes are limited by 3 points and 3 segments connecting these points in pairs.
All vertices of any triangle, regardless of its type, are designated by capital letters in Latin letters, and its sides are depicted by the corresponding designations of opposite vertices, but not in capital letters, 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 will end up with external corner. 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 background
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 XV – 16th centuries They began to conduct a lot of research on the properties of a triangle, and as a result such a science 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.
Which one is the best? famous triangle You know? This is of course the Bermuda Triangle! It received this name in the 50s because of the geographical location of the points (vertices of the triangle), within which, according to the existing theory, anomalies associated with it arose. The vertices of the Bermuda Triangle are Bermuda, Florida and Puerto Rico.
Assignment: What theories about Bermuda Triangle did you hear?
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, capital letter And 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?
Today we are going to the country of Geometry, where we will get acquainted with different types of triangles.
Consider the geometric shapes 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 party - 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 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.
References
- 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: Methodical recommendations 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.
When studying mathematics, students begin to become familiar with various types of geometric shapes. Today we will talk about various types triangles.
Definition
Geometric figures that consist of three points that are not on the same line are called triangles.
The segments connecting the points are called sides, and the points are called vertices. Vertices are designated in capital letters, for example: A, B, C.
The sides are designated by the names of the two points from 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 by angles and sides. Each type of triangle has its own properties.
There are three types of triangles at the corners:
- acute-angled;
- rectangular;
- obtuse-angled.
All angles acute-angled triangles are acute, that is degree measure each is no more than 90 0.
Rectangular a triangle contains a right angle. The other two angles will always be acute, since otherwise the sum of the angles of the triangle will exceed 180 degrees, and this is impossible. The side that is opposite the right angle is called the hypotenuse, and the other two are called the legs. The hypotenuse is always larger 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 at 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 make simple tasks in geometry. Therefore, remember: if two sides of a triangle are equal to 3, then the third 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.
Isosceles triangle - a triangle with only two sides equal. These sides are called lateral, and the third is called 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 the problem does not contain any clarifications about the figure, 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 angles of a triangle, regardless of its type, is 1800.
Against larger angle there is a big 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 the golden triangle. This is an isosceles triangle in which 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.
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 according to their sides and the size of their angles. Triangles have certain properties that can be used to solve problems.
