What does segment mean? Dot

Straight

The concept of a straight line, as well as the concept of a point, are the basic concepts of geometry. As you know, the basic concepts are not defined. This is no exception to the concept of a straight line. Therefore, let us consider the essence of this concept through its construction.

Let's take a ruler and, without lifting the pencil, draw a line of arbitrary length (Fig. 1).

We will call the resulting line direct. However, it should be noted here that this is not the entire straight line, but only part of it. It is not possible to construct the entire straight line; it is infinite at both its ends.

We will denote straight lines by a small Latin letter or its two dots in parentheses (Fig. 2).

The concepts of a straight line and a point are connected by three axioms of geometry:

Axiom 1: For every arbitrary line there are at least two points that lie on it.

Axiom 2: You can find at least three points that do not lie on the same line.

Axiom 3: In $2$ arbitrary points there is always a straight line, and this straight line is unique.

For two straight lines it is relevant relative position. Three cases are possible:

1. Two straight lines coincide. In this case, each point of one line will also be a point of the other line.
2. Two lines intersect. In this case, only one point from one line will also belong to the other line.
3. Two lines are parallel. In this case, each of these lines has its own set of points that are different from each other.

In this article we will not dwell on these concepts in detail.

Segment

Let us be given an arbitrary straight line and two points belonging to it. Then

Definition 1

A segment will be called a part of a line that is bounded by two of its arbitrary distinct points.

Definition 2

The points that limit a segment within the framework of Definition 1 are called the ends of this segment.

We will denote the segments by its two end points in square brackets (Fig. 3).

Comparison of segments

Let's consider two arbitrary segments. Obviously, they can be either equal or unequal. To understand this, we need the following axiom of geometry.

Axiom 4: If both ends of two different segments coincide when they are superimposed, then such segments will be equal.

So, to compare the segments we have chosen (let’s denote them as segment 1 and segment 2), we will superimpose the end of segment 1 onto the end of segment 2, so that the segments remain on one side of these ends. After such an overlay, the following two cases are possible:

Section length

In addition to comparing one segment with another, measuring segments is also often necessary. To measure a segment means to find its length. To do this, you need to select some kind of “reference” segment, which we will take as a unit (for example, a segment whose length is 1 centimeter). After selecting such a segment, we compare the segments with it, the length of which needs to be found. Let's look at an example.

Example 1

Find the length of the next segment

if the next segment equals 1

To measure it, let’s take the segment $$ as a standard. We will postpone it for the segment $$. We get:

Answer: $6$ see

The concept of the length of a segment is associated with the following axioms of geometry:

Axiom 5: By choosing a certain unit of measurement for segments, the length of any segment will be positive.

Axiom 6: By choosing a certain unit of measurement for segments, for any positive number we can find a segment whose length is equal to the given number.

After determining the length of the segments, we have a second way to compare segments. If, with the same choice of unit of length, the segment $1$ and the segment $2$ have the same length, then such segments will be called equal. If, without loss of generality, segment 1 has a length of numerical value is less than the length of the segment $2$, then the segment $1$ will be less than the segment $2$.

The most in a simple way Measuring the length of segments is measuring using a ruler.

Example 2

Write down the lengths of the following segments:

Let's measure them using a ruler:

1. $4$ see
2. $10$ see
3. $5$ see
4. $8$ see

We usually hear the word segment when talking about geometry or mathematical analysis. In both areas given word denotes very similar concepts, namely the part of a line that is bounded by two points.

A segment in everyday life

Of course, we hear the word “segment” not only when discussing mathematical issues; it is also used in everyday speech. So what is a segment in the everyday sense of the word? As a rule, when pronouncing the word “segment”, a person means a piece of this or that material that needs to be cut from something. For example, we may need a piece of fabric, a piece of tape, a piece of tape, and much more.

Segment in mathematics

As we said above, in mathematics, a segment is a part of a line bounded by two points, but sometimes you can also come across such a term - a set of numbers or points on a line between two numbers or points. It sounds much more scientific and complex, but if you think about it, both definitions mean the same thing.

Other meanings

The word “segment” is also pronounced when they want to designate the passage of a certain stage, for example, “segment of a path” or “segment of time.” You've probably seen such phrases in books.

In addition, the period after the abolition of serfdom in Russia was called land plots, which were seized by landowners from peasants.

These are the definitions of the word “segment”. Find out the meanings of new words in the section.

Segment. Length of the segment. Triangle.

1. In this paragraph you will be introduced to some concepts of geometry. Geometry- the science of "measuring the earth." This word comes from the Latin words: geo - earth and metr - measure, to measure. In geometry, various geometric objects, their properties, their connections with the outside world. The simplest geometric objects are a point, a line, a surface. More complex geometric objects, e.g. geometric shapes and bodies formed from protozoa.

If we apply a ruler to two points A and B and draw a line along it connecting these points, we get segment, which is called AB or VA (we read: “a-be”, “be-a”). Points A and B are called ends of the segment(Figure 1). The distance between the ends of a segment, measured in units of length, is called lengthcutka.

Units of length: m - meter, cm - centimeter, dm - decimeter, mm - millimeter, km - kilometer, etc. (1 km = 1000 m; 1 m = 10 dm; 1 dm = 10 cm; 1 cm = 10 mm). To measure the length of segments, use a ruler or tape measure. To measure the length of a segment means to find out how many times a particular length measure fits into it.

Equal are called two segments that can be combined by superimposing one on the other (Figure 2). For example, you can actually or mentally cut out one of the segments and attach it to another so that their ends coincide. If the segments AB and SK are equal, then we write AB = SK. Equal segments have equal lengths. The opposite is true: two segments of equal length are equal. If two segments have different lengths, then they are not equal. Of two unequal segments, the smaller one is the one that forms part of the other segment. You can compare overlapping segments using a compass.

If we mentally extend the segment AB in both directions to infinity, then we will get an idea of direct AB (Figure 3). Any point lying on a line splits it into two beam(Figure 4). Point C splits line AB into two beam SA and SV. Tosca C is called the beginning of the ray.

2. If three points that do not lie on the same line are connected by segments, then we get a figure called triangle. These points are called peaks triangle, and the segments connecting them are parties triangle (Figure 5). FNM - triangle, segments FN, NM, FM - sides of the triangle, points F, N, M - vertices of the triangle. The sides of all triangles have the following property: d The length of any side of a triangle is always less than the sum of the lengths of its other two sides.

If you mentally extend, for example, the surface of a table top in all directions, you will get an idea of plane. Points, segments, straight lines, rays are located on a plane (Figure 6).

Block 1. Additional

The world in which we live, everything that surrounds us, the ancients called nature or space. The space in which we live is considered three-dimensional, i.e. has three dimensions. They are often called: length, width and height (for example, the length of a room is 4 m, the width of a room is 2 m and the height is 3 m).

The idea of ​​a geometric (mathematical) point is given to us by a star in the night sky, a dot at the end of this sentence, a mark from a needle, etc. However, all of the listed objects have dimensions; in contrast, the dimensions of a geometric point are considered equal to zero (its dimensions are equal to zero). Therefore, a real mathematical point can only be imagined mentally. You can also tell where it is located. By placing a dot in a notebook with a fountain pen, we will not depict a geometric point, but we will assume that the constructed object is a geometric point (Figure 6). The dots indicate in capital letters Latin alphabet: A, B, C, D, (read " point a, point be, point tse, point de") (Figure 7).

Wires hanging on poles, a visible horizon line (the boundary between sky and earth or water), a riverbed depicted on a map, a gymnastics hoop, a stream of water gushing from a fountain give us an idea of ​​lines.

There are closed and open lines, smooth and non-smooth lines, lines with and without self-intersection (Figures 8 and 9).

A sheet of paper, laser disc, soccer ball shell, packaging box cardboard, Christmas plastic mask, etc. give us an idea of surfaces(Figure 10). When painting the floor of a room or a car, the surface of the floor or car is covered with paint.

Human body, stone, brick, cheese, ball, ice icicle, etc. give us an idea of geometric bodies (Figure 11).

The simplest of all lines is it's straight. Place a ruler on a sheet of paper and draw a straight line along it with a pencil. Mentally extending this line to infinity in both directions, we will get the idea of ​​a straight line. It is believed that a straight line has one dimension - length, and its other two dimensions are equal to zero (Figure 12).

When solving problems, a straight line is depicted as a line that is drawn along a ruler with a pencil or chalk. Direct lines are designated by lowercase Latin letters: a, b, n, m (Figure 13). You can also denote a straight line by two letters corresponding to the points lying on it. For example, straight n in Figure 13 we can denote: AB or VA, ADorDA,DB or BD.

Points can lie on a line (belong to a line) or not lie on a line (not belong to a line). Figure 13 shows points A, D, B lying on line AB (belonging to line AB). At the same time they write. Read: point A belongs to line AB, point B belongs to AB, point D belongs to AB. Point D also belongs to line m, it is called general dot. At point D the lines AB and m intersect. Points P and R do not belong to straight lines AB and m:

Through any two points always you can draw a straight line and only one .

Of all types of lines connecting any two points, the segment whose ends are these points has the shortest length (Figure 14).

A figure that consists of points and segments connecting them is called a broken line (Figure 15). The segments that form a broken line are called links broken line, and their ends - peaks broken line A broken line is named (designated) by listing all its vertices in order, for example, the broken line ABCDEFG. The length of a broken line is the sum of the lengths of its links. This means that the length of the broken line ABCDEFG is equal to the sum: AB + BC + CD + DE + EF + FG.

A closed broken line is called polygon, its vertices are called vertices of the polygon, and its links parties polygon (Figure 16). A polygon is named (designated) by listing in order all its vertices, starting from any one, for example, polygon (heptagon) ABCDEFG, polygon (pentagon) RTPKL:

The sum of the lengths of all sides of a polygon is called perimeter polygon and is denoted by the Latin letterp(read: pe). Perimeters of polygons in Figure 13:

P ABCDEFG = AB + BC + CD + DE + EF + FG + GA.

P RTPKL = RT + TP + PK + KL + LR.

Mentally extending the surface of a table top or window glass to infinity in all directions, we get an idea of ​​the surface, which is called plane (Figure 17). The planes are designated in small letters of the Greek alphabet: α, β, γ, δ, ... (we read: plane alpha, beta, gamma, delta, etc.).

Block 2. Vocabulary.

Make a dictionary of new terms and definitions from §2. To do this, enter words from the list of terms below in the empty rows of the table. In Table 2, indicate the term numbers in accordance with the line numbers. It is recommended that you carefully review §2 and block 2.1 before filling out the dictionary.

Block 3. Establish correspondence (CS).

Geometric shapes.

Block 4. Self-test.

Measuring a segment using a ruler.

Let us recall that to measure a segment AB in centimeters means to compare it with a segment 1 cm long and find out how many such 1 cm segments fit in the segment AB. To measure a segment in other units of length, proceed in the same way.

To complete the tasks, work according to the plan given in the left column of the table. In this case, we recommend covering the right column with a sheet of paper. You can then compare your findings with the solutions in the table to the right.

Block 5. Establishing a sequence of actions (SE).

Constructing a segment of a given length.

Option 1. The table contains a mixed up algorithm (a mixed up order of actions) for constructing a segment of a given length (for example, let’s build a segment BC = 7 cm). In the left column is an indication of the action, in the right column is the result of this action. Rearrange the rows of the table so that you get the correct algorithm for constructing a segment of a given length. Write down the correct sequence of actions.

Option 2. The following table shows the algorithm for constructing the segment KM = n cm, where instead of n You can substitute any number. In this option there is no correspondence between action and result. Therefore, it is necessary to establish a sequence of actions, then for each action, select its result. Write the answer in the form: 2a, 1c, 4b, etc.

Option 3. Using the algorithm of option 2, construct segments in your notebook at n = 3 cm, n = 10 cm, n = 12 cm.

Block 6. Facet test.

Segment, ray, straight line, plane.

In the tasks of the facet test, pictures and records numbered 1 - 12, given in Table 1, are used. Task data is formed from them. Then the requirements of the tasks are added to them, which are placed in the test after the connecting word “TO”. Answers to the problems are placed after the word “EQUAL”. The set of tasks is given in Table 2. For example, task 6.15.19 is composed as follows: “IF the problem uses Figure 6 , s Then condition number 15 is added to it, the task requirement is number 19.”

13) construct four points so that every three of them do not lie on the same straight line;

14) draw a straight line through every two points;

15) mentally extend each of the surfaces of the box in all directions to infinity;

16) the number of different segments in the figure;

17) the number of different rays in the figure;

18) the number of different straight lines in the figure;

19) the number of different planes obtained;

20) length of segment AC in centimeters;

21) length of segment AB in kilometers;

22) length of segment DC in meters;

23) perimeter of triangle PRQ;

24) length of the broken line QPRMN;

25) quotient of the perimeters of triangles RMN and PRQ;

26) length of segment ED;

27) length of segment BE;

28) the number of resulting points of intersection of lines;

29) the number of resulting triangles;

30) the number of parts into which the plane was divided;

31) the perimeter of the polygon, expressed in meters;

32) the perimeter of the polygon, expressed in decimeters;

33) the perimeter of the polygon, expressed in centimeters;

34) the perimeter of the polygon, expressed in millimeters;

35) perimeter of the polygon, expressed in kilometers;

EQUAL (equal, has the form):

a) 70; b) 4; c) 217; d) 8; e) 20; e) 10; g) 8∙b; h) 800∙b; i) 8000∙b; j) 80∙b; k) 63000; m) 63; m) 63000000; o) 3; n) 6; p) 630000; c) 6300000; t) 7; y) 5; t) 22; x) 28

Block 7. Let's play.

7.1. Math labyrinth.

The labyrinth consists of ten rooms with three doors each. In each of the rooms there is one geometric object (it is drawn on the wall of the room). Information about this object is in the “guide” to the labyrinth. While reading it, you need to go to the room that is written about in the guidebook. As you walk through the rooms of the labyrinth, draw your route. The last two rooms have exits.

Guide to the Labyrinth

1. You must enter the labyrinth through a room where there is a geometric object that has no beginning, but has two ends.
2. The geometric object of this room has no dimensions, it is like a distant star in the night sky.
3. The geometric object of this room is composed of four segments that have three common points.
4. This geometric object consists of four segments with four common points.
5. This room contains geometric objects, each of which has a beginning but no end.
6. Here are two geometric objects that have neither beginning nor end, but with one common point.

1. An idea of ​​this geometric object is given by the flight of artillery shells

(trajectory of movement).

1. This room contains a geometric object with three peaks, but they are not mountainous.

1. The flight of a boomerang gives an idea of ​​this geometric object (hunting

weapons of the indigenous people of Australia). In physics this line is called a trajectory

body movements.

1. An idea of ​​this geometric object is given by the surface of the lake in

calm weather.

Now you can exit the maze.

The maze contains geometric objects: plane, open line, straight line, triangle, point, closed line, broken line, segment, ray, quadrilateral.

7.2. Perimeter of geometric shapes.

In the drawings, highlight geometric shapes: triangles, quadrangles, pentagons and hexagons. Using a ruler (in millimeters), determine the perimeters of some of them.

7.3. Relay race of geometric objects.

Relay tasks have empty frames. Write down the missing word in them. Then move this word to another frame where the arrow points. In this case, you can change the case of this word. As you go through the stages of the relay, complete the required formations. If you complete the relay correctly, you will receive the following word at the end: perimeter.

7.4. Strength of geometric objects.

Read § 2, write down the names of geometric objects from its text. Then write these words in the empty cells of the “fortress”.

>>Mathematics 7th grade. Complete lessons >>Geometry: Line segment. Complete lessons

Segment

A segment is a part of a line that contains two different points A and B of this line (the ends of the segment) and all the points of the line that lie between them (the internal points of the segment).

Straight segment is a set (part of a line) consisting of two different points and all the points lying between them. A straight line segment connecting two points A and B (which are called the ends of the segment) is denoted as follows -. If square brackets are omitted in the designation of a segment, then write “segment AB”. Any point lying between the ends of a segment is called its interior point. The distance between the ends of a segment is called its length and is denoted as |AB|.

To denote a segment with ends at points A and B, we will use the symbol.

About a point C belonging to a segment AB, we also say that point C lies between points A and B (if C is an internal point of the segment), and also that segment AB contains point C.

The property of a segment is given by the axiom:

Axiom:
Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its internal points. AB = AC + CB.

The distance between two points A and B is called segment length AB.
Moreover, if points A and B coincide, we will assume that the distance between them is zero.
Two segments are called equal if their lengths are equal.

Segment AC=DE, CB=EF And AB=DF

On Figure 1 shows a line a and 3 points on this line: A, B, C. Point B lies between points A and C, we can say that it separates points A and C. Points A and C lie on opposite sides of point B. Points B and C are located on one side of point A, points A and B are on the same side of point C.

figure 1

Segment- part of a line, which consists of all points of this line lying between these points, which are called the ends of the segment. A segment is indicated by indicating its end points. When they say segment AB, they mean a segment with ends at points A and B.

At this point Figure 2 we see segment AB, it is part of a line. Point X lies between points A and B, so it belongs to the segment AB, point Y does not lie between points A and B, so it does not belong to the segment AB.

figure 2

The main property of the location of points on a line is that out of three points on a line, only one lies between two points.

Point A lies between X and Y.

Point X divides segment AB.

Usually, for a straight line segment, it does not matter in what order its ends are considered: that is, segments AB and BA represent the same segment. If the segment has direction, that is, the order in which its ends are listed, then such a segment is called directed. For example, the above directed segments do not coincide. There is no special designation for directed segments - the fact that the segment is important and its direction is usually indicated separately.

Further generalization leads to the concept vector- the class of all equal in length and codirectional directed segments.

Crossword

1. The pen moves along the sheet. Along the line, along the edge. It turns out the trait is called...
2. Ancient Greek scientist.
3. The result of an instant touch.
4. A textbook consisting of 13 volumes, which for many centuries was the main guide to geometry.
5. Ancient Greek scientist, author of the collective work “Principles”.
6. Unit of length.
7. A part of a line bounded by two points.
8. Unit of measurement of length in Ancient Egypt.
9. An ancient Greek mathematician who proved the theorem that bears his name.
10. Є mathematical sign.
11. Geometry section.

Interesting fact:

In geometry, paper is used to: write, draw; cut; bend. The subject of mathematics is such a serious subject that it is good to take every opportunity to make it a little fun.

Crop circles are an intergalactic language of communication between alien intelligent beings
Crop circles... There are so many different opinions, so many fortune-telling, so many hypotheses, but there are no intelligible explanations of what it is.
Crop circles... They fascinate people with their laconic beauty, they irritate us with their incomprehensibility of origin and purpose.

Questions:

1) What is a segment?

2) What is the length of the segment?

3) Difference between a segment and a vector?

List of sources used:

1. Program for educational institutions. Mathematics. Ministry of Education of the Russian Federation.
2. Federal general educational standard. Bulletin of Education. No. 12, 2004.
3. Programs of general education institutions. Geometry grades 7-9. Authors: S.A. Burmistrova. Moscow. "Enlightenment", 2009.
4. Kiselev A.P. "Geometry" (planimetry, stereometry)

Edited and sent by Poturnak S.A.