The science of geometry tells us about what a triangle, square, cube is. In the modern world, it is studied in schools by everyone, without exception. Also, a science that studies directly what a triangle is and what properties it has is trigonometry. She examines in detail all the phenomena associated with data.We will talk about what a triangle is today in our article. Below will be described their types, as well as some theorems associated with them.
What is a triangle? Definition
It is a flat polygon. It has three corners, which is clear from its name. It also has three sides and three vertices, the first of which are line 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 180.
What are triangles?
They can be classified according to various criteria.
First of all, they are divided into acute-angled, obtuse-angled and rectangular. The first have sharp corners, that is, those that are less than 90 degrees. In obtuse angles, one of the corners is obtuse, that is, one that is more than 90 degrees, the other two are sharp. TO acute-angled triangles are also equilateral. For such triangles, all sides and angles are 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 line), that is, two of its sides are perpendicular. The other two corners are sharp. They can be equal, then it will be isosceles. The Pythagorean theorem is associated with a right-angled triangle. With the help of 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 remember about an isosceles triangle. This is one in which two of the sides are equal, and the two angles are also equal.
What are leg and hypotenuse?
A leg is one of the sides of a triangle that form an angle of 90 degrees. The hypotenuse is the remaining side that is opposite right angle... From it, a perpendicular can be lowered onto the leg. The ratio of the adjacent leg to the hypotenuse is called cosine, and the opposite is called sine.
- what are its features?
It is rectangular. Its legs are three and four, and the hypotenuse is five. If you saw that the legs of this triangle are equal to three and four, you can rest assured that the hypotenuse will be equal to five. Also, according to 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 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 5. Also, the Egyptian triangle is called a right-angled triangle, the sides of which are 6, 8 and 10; 9, 12 and 15 and other numbers with a ratio of 3: 4: 5.
What else could a triangle be?
Also, triangles can be inscribed and described. The figure around which the circle is described is called inscribed, all its vertices are points lying on the circle. The described triangle is the one in which the circle is inscribed. All its sides are in contact with it at certain points.
How is
The area of any figure is measured in square units(square meters, square millimeters, square centimeters, square decimeters, etc.) This value can be calculated in various ways, depending on the type of triangle. The area of any figure with corners can be found by multiplying its side by the perpendicular dropped on it from opposite corner, and dividing this figure by two. You can also find this value by multiplying the two sides. Then multiply this number by the sine of the angle between the given sides, and divide this result by two. Knowing all the sides of the triangle, but not knowing its angles, you can find the area in another way. To do this, you need to find half of the perimeter. Then subtract one by one from the given 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 sides and dividing the resulting number by which is described around it, multiplied by four.
The area of the described triangle is found in this way: we multiply half of the perimeter by the radius of the circle that is inscribed in it. If then its area can be found as follows: we 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, for this one of them must be multiplied by the root of three, and then this number must be divided by two.
Triangle theorems
The main theorems associated with this figure are the Pythagorean theorem described above and cosines. The second (sine) is that if you divide any side by the sine of its opposite angle, you can get the radius of the circle that is described around it, multiplied by two. The third (cosines) is that if you subtract their product, multiplied by two and by the cosine of the angle between them, from the sum of the squares of the two sides, you get the square of the third side.
Dali triangle - what is it?
Many, 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 triangle is common name three places that are closely associated with the life of the famous artist. Its "peaks" are the house in which Salvador Dali lived, the castle, which he gave to his wife, and the museum of surrealist paintings. During a tour of these places, you can learn a lot interesting facts about this kind of creative artist known all over the world.
Triangle - definition and general concepts
A triangle is a simple polygon with three sides and the same number of angles. Its planes are limited by 3 points and 3 line segments connecting these points in pairs.
All vertices of any triangle, regardless of its type, are designated in capital with Latin letters, and its sides are depicted by the corresponding designations of opposite vertices, but not in big letters, but small. So, for example, a triangle with vertices designated by the letters A, B and C has sides a, b, c.
If we consider a triangle in Euclidean space, then this is such a geometric figure that was formed with the help of three segments connecting three points that do not lie on one straight line.
Look closely at the picture 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 its corners inside.
Types of triangles
According to the size, angles of triangles, they are divided into such varieties as: Rectangular;
Acute-angled;
Obtuse.
Rectangular triangles are those that have one right angle, and the other two have acute angles.
Acute triangles are those in which all of its corners are sharp.
And if a triangle has one obtuse angle, and the other two angles are acute, then such a triangle is classified as obtuse.
Each of you understands perfectly well that not all triangles have equal sides. And according to how long its sides have, the triangles can be divided into:
Isosceles;
Equilateral;
Versatile.
Assignment: Draw different types triangles. Give them a definition. What difference do you see between them?
Basic properties of triangles
Although these simple polygons may differ from each other in the magnitude of the angles or sides, each triangle has basic properties that are characteristic of this figure.
In any triangle:
The total sum of all its angles is 180º.
If it belongs to the equilateral, then each of its angles is 60º.
An equilateral triangle has the same and even angles to each other.
The smaller the side of the polygon, the smaller the angle is opposite to it, and vice versa, opposite the larger side is the larger angle.
If the sides are equal, then equal angles are located opposite them, and vice versa.
If we take a triangle and extend its side, then in the end we are formed outer corner... He is equal to the sum inner 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 into the table:
1. How many degrees does the third angle have?
2. What kind of triangles does it belong to?
Signs of equality of triangles
I sign
II sign
III sign
Height, bisector, and median of a triangle
Height of a triangle - the perpendicular drawn from the top of the figure to its opposite side is called the height of the triangle. All the heights of the triangle intersect at one point. The point of intersection of all 3 heights of the triangle is its orthocenter.
The segment drawn from this vertex and connecting it in the middle of the opposite side is the median. The medians, as well as the heights of the 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 the 2 sides of the triangle is called the midline.
History reference
A figure such as a triangle has been known since 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 the triangle moved to more high level but still, it happened over two thousand years ago.
In XV - XVI centuries began to conduct a lot of research on the properties of the triangle and as a result there was such a science as planimetry, which was called "New geometry of the triangle."
A scientist from Russia N.I. Lobachevsky made a huge contribution to the knowledge of the properties of triangles. His works later found application both in mathematics and physics and cybernetics.
Thanks to the 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 application is simply necessary in drawing up maps, measuring areas, and in the design of various mechanisms.
What is the most 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 (the vertices of the triangle), within which, according to the existing theory, anomalies associated with it arose. The peaks of the Bermuda Triangle are Bermuda, Florida and Puerto Rico.
Assignment: What theories about 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 the geometry of Riemann, the sum of all the angles of a triangle is greater than 180 degrees, and in the writings 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, which was drawn from the apex 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. What is a triangle whose two sides are equal?
4. What is the name of a triangle that has an angle of 90 °?
5. What is the name of the large side of the triangle?
6. 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. Name of the line segment connecting the top of our shape with the middle of the opposite side?
10. In a simple polygon ABC, capital letter Is it ...?
11. What is the name of the segment dividing the angle of the triangle in half.
Questions about triangles:
1. Give a definition.
2. How many heights does it have?
3. How many bisectors does a triangle have?
4. What is the sum of its angles?
5. What kinds of this simple polygon do you know?
6. What are the points of the triangles that are called wonderful?
7. What device can be used to measure the angle?
8. If the hands of the clock show 21 o'clock. What is the angle of the hour hands?
9. At what angle does the person turn if he is given the command "to the left", "around"?
10. What other definitions do you know that are associated with a figure with three corners and three sides?
Subjects> Mathematics> Grade 7 Mathematics
Today we go to the country of Geometry, where we will get acquainted with different types of triangles.
Consider the geometric shapes and find among them "superfluous" (Fig. 1).
Rice. 1. Illustration for example
We see that the figures # 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 one straight line, and three segments that connect these points in pairs.
The points are called the vertices of the triangle, segments - it parties... The sides of the triangle form there are three corners at the vertices of the triangle.
The main signs of a triangle are three sides and three corners. In terms of angle, triangles are acute-angled, rectangular and obtuse-angled.
A triangle is called acute-angled if all three corners are acute, that is, less than 90 ° (Fig. 4).
Rice. 4. Acute-angled triangle
A triangle is called rectangular if one of its corners is 90 ° (Fig. 5).
Rice. 5. Right-angled triangle
A triangle is called obtuse if one of its corners is obtuse, that is, more than 90 ° (Fig. 6).
Rice. 6. Obtuse triangle
By number equal sides triangles are equilateral, isosceles, versatile.
An isosceles triangle is a triangle whose two sides are equal (Fig. 7).
Rice. 7. Isosceles triangle
These parties are called lateral, the third side - basis. In an isosceles triangle, the angles at the base are equal.
Isosceles triangles are acute-angled and obtuse-angled(fig. 8) .
Rice. 8. Acute and obtuse isosceles triangles
An equilateral triangle is a triangle 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. Versatile triangle
Complete the task. Divide these triangles into three groups (fig. 11).
Rice. 11. Illustration for the task
First, we distribute by the magnitude of the angles.
Acute triangles: No. 1, No. 3.
Rectangular 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.
Versatile triangles: No. 4, No. 6.
Isosceles triangles: No. 2, No. 3, No. 5.
Equilateral triangle: No. 1.
Consider the drawings.
Think about which piece of wire you made each triangle (fig. 12).
Rice. 12. Illustration for the task
You can reason like this.
The first piece of wire is divided into three equal parts, so an equilateral triangle can be made from it. In the figure, he is shown as the third.
The second piece of wire is divided into three different parts, so it can be made from versatile triangle... He is shown first in the figure.
The third piece of wire is divided into three parts, where the two parts are the same length, which means that an isosceles triangle can be made from it. In the figure, he is shown as the second.
Today in the lesson we got acquainted with the different types of triangles.
Bibliography
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Education", 2012.
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Education", 2012.
- M.I. Moreau. Mathematics lessons: Guidelines for the teacher. Grade 3. - M .: Education, 2012.
- Normative legal document. Monitoring and evaluation of learning outcomes. - M .: "Education", 2011.
- "School of Russia": Programs for primary school... - M .: "Education", 2011.
- S.I. Volkova. Mathematics: Verification work. Grade 3. - M .: Education, 2012.
- V.N. Rudnitskaya. Tests. - M .: "Exam", 2012.
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Homework
1. Complete the phrases.
a) A triangle is a figure that consists of ..., not lying on one straight line, and ..., connecting these points in pairs.
b) Points are called …, segments - it …... The sides of the triangle form at the vertices of the triangle ….
c) In terms of angle, triangles are…,…,….
d) According to the number of equal sides, triangles are…,…,….
2. Draw
a) a right-angled triangle;
b) acute-angled triangle;
c) obtuse triangle;
d) an equilateral triangle;
e) versatile triangle;
f) isosceles triangle.
3. Make an assignment on the topic of the lesson for your peers.
When studying mathematics, students begin to become familiar with different types of geometric shapes... Today we will talk about different types triangles.
Definition
Geometric shapes that are made up of three points that are not on the same straight line are called triangles.
The lines that connect the points are called sides, and the points are called vertices. Vertices are indicated by capital Latin letters, for example: A, B, C.
The sides are designated by the names of the two points of which they are composed - AB, BC, AC. Crossing, the sides form corners. 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 corner triangles:
- acute-angled;
- rectangular;
- obtuse.
All angles acute-angled triangles are sharp, 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 sharp, 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 larger than the leg.
Obtuse a triangle contains an obtuse angle. That is, an angle greater than 90 degrees. The other two corners in such a triangle will be sharp.
Rice. 2. Types of triangles in the corners.
A Pythagorean triangle is a rectangle whose sides are equal to 3, 4, 5.
Moreover, the big side is the hypotenuse.
Such triangles are often used to compose simple tasks in geometry. Therefore, remember: if the two sides of the triangle are equal to 3, then the third will necessarily 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 triangle - a triangle in which only two sides are equal. These sides are called side, and the third is called the base. In addition, the angles at the base of an isosceles triangle are equal and always sharp.
Versatile or an arbitrary triangle is a triangle in which all lengths and all angles are not equal to each other.
If in the problem there are no clarifications about the figure, then it is considered 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.
Against greater angle there is a big side. And also the length of any side is always less than the sum of its two other sides. These properties are confirmed by the triangle inequality theorem.
There is the 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 of 2: 2: 1.
Task:
Is there a triangle whose sides are 6 cm, 3 cm, 4 cm?
Solution:
To solve this problem, 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 to solve problems.
