What is a cut? Dot. Line segment. Ray. Straight. Number line 2 what is a segment

Line segment. Cut length. Triangle.

1. In this paragraph, you will get acquainted with 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 surrounding world. The simplest geometric objects are a point, a line, a surface. More complex geometric objects, such as geometric shapes and bodies, are formed from the simplest ones.

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

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

Equal two segments are called, which can be combined by superimposing one on the other (Figure 2). For example, one can cut out one of the segments, actually or mentally, and attach it to another so that their ends coincide. If the segments AB and SK are equal, then write AB = SK. Equal segments have equal lengths. The converse 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 segments by superposition using a compass.

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

2. If three points that do not lie on one straight line are connected by segments, then we get a figure called triangle. These points are called peaks triangles, and the segments connecting them, 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: The length of any side of a triangle is always less than the sum of the lengths of the other two sides.

If we mentally extend in all directions, for example, the surface of the table top, we 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 to be three-dimensional, i.e. has three dimensions. They are often called: length, width and height (for example, the length of the room is 4 m, the width of the 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 trace from a needle, etc. However, all the listed objects have dimensions, in contrast to them, 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 mentally represented. You can also tell where it is. Putting a point 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). Points are denoted by capital letters of the Latin alphabet: A, B, C, D, (read " dot a, dot be, dot ce, dot de") (Figure 7).

Wires hanging on poles, the visible horizon line (the border between heaven and earth or water), the riverbed shown on the map, the gymnastic hoop, the stream of water spouting from the fountain give us an idea of ​​the lines.

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

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

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

The simplest of all lines - it's straight. We will attach a ruler to a sheet of paper and draw a straight line along it with a pencil. Mentally continuing this line to infinity in both directions, we get an idea of ​​a straight line. It is believed that the straight line has one dimension - the 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. Straight lines are indicated by lowercase Latin letters: a, b, n, m (Figure 13). A line can also be denoted by two letters corresponding to points lying on it. For example, straight n Figure 13 shows: AB or BA, ADorDBUT,DB or BD.

Points can lie on a line (belong to a line) and 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. The point D also belongs to the line m, it is called general dot. At point D, lines AB and m intersect. Points P and R do not belong to lines AB and m:

Through any two points always it is possible to draw a straight line, and moreover, only one .

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

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

A closed broken line is called polygon, its vertices are called polygon vertices, and its links parties polygon (Figure 16). They name (designate) a polygon, listing in order all its vertices, starting with any, for example, polygon (septagon) 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). The perimeters of the 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 denoted by small letters of the Greek alphabet: α, β, γ, δ, ... (read: plane alpha, beta, gamma, delta, etc.).

Block 2. Dictionary.

Compile a glossary of new terms and definitions from §2. To do this, in the empty rows of the table, enter the words from the list of terms below. In table 2, indicate the number of terms in accordance with the line numbers. It is recommended to carefully review §2 and block 2.1 before completing the dictionary.

Block 3. Establish a match (CA).

Geometric figures.

Block 4. Self-test.

Measuring a line with a ruler.

Recall that to measure the 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 a similar way.

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

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

Construction of a segment of a given length.

Option 1. The table contains a confused algorithm (a confused order of actions) for constructing a segment of a given length (for example, we construct a segment BC = 7cm). In the left column, an indication of the action; in the right column, the result of performing 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 any number can be substituted. In this variant 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 down the answer in the form: 2a, 1c, 4b, etc.

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

Block 6. Facet test.

Segment, ray, line, plane.

In the tasks of the facet test, figures and records numbered 1 - 12 are used, given in Table 1. From them, task data is formed. Then the requirements of the tasks are added to them, which are placed in the test after the connecting word "TO". Answers to the tasks 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 task uses Figure 6 , h Then the 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 one 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 lines in the figure;

19) the number of resulting different planes;

20) length of segment AC in centimeters;

21) the length of the segment AB in kilometers;

22) length of segment DC in meters;

23) the perimeter of the triangle PRQ;

24) the length of the polyline QPRMN;

25) the 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) the 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; r) 7; y) 5; f) 22; x) 28

Block 7. Let's play.

7.1. Mathematical maze.

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. Reading it, you must go to the room, which is written in the guide. Passing through the rooms of the labyrinth, draw your route. The last two rooms have exits.

maze guide

1. You must enter the labyrinth through the 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 made up of four segments that have three common points.
4. This geometric object consists of four segments with four common points.
5. In this room are 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. The 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 vertices, but these are not mountain

1. The flight of a boomerang (hunting

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

body movements.

1. The idea of ​​this geometric object gives the surface of the lake in

windless weather.

Now you can exit the maze.

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

7.2. The perimeter of geometric shapes.

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

7.3. Relay race of geometric objects.

The tasks of the relay 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. Passing through the stages of the relay, perform the required constructions. If you pass the relay correctly, then at the end you will receive the word: perimeter.

7.4. Fortress of geometric objects.

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

>>Mathematics Grade 7. Full lessons >>Geometry: Line segment. Complete Lessons

Line 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 points of the line that lie between them (the interior points of the segment).

Line segment is a set (part of a line) consisting of two different points and all points lying between them. A 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 “segment AB” is written. 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 .

A point C belonging to segment AB is also said to lie between points A and B (if C is an interior 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 interior points. AB=AC+CB.

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

Line segment AC=DE, CB=EF and AB=DF

On the figure 1 a line a and 3 points on this line are shown: 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 on the same side of point A, points A and B are on the same side of point C.

picture 1

Line segment- a part of a line, which consists of all points of this line, lying between the given points, which are called the ends of the segment. A line segment is denoted by specifying its endpoints. When they say segment AB, t means a segment with ends at points A and B.

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

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

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

Crossword

1. The pen goes along the sheet. Along the line, along the edge. It turns out the feature is called ...
2. Ancient Greek scientist.
3. Instant touch result.
4. An educational book consisting of 13 volumes, which for many centuries was the main guide to geometry.
5. Ancient Greek scientist, author of the collective work "Beginnings".
6. Unit of measure for length.
7. The part of a line bounded by two points.
8. A unit of length in ancient Egypt.
9. 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 so serious that it is useful not to miss opportunities to make it a little entertaining.

Crop circles - intergalactic language of communication of alien intelligent beings
Crop circles ... How many different opinions, how many fortune-telling, how many hypotheses, but there are no intelligible explanations of what it is.
Crop circles ... They fascinate people with their laconic beauty, they annoy us with their dullness of origin and destination.

Questions:

1) What is a segment?

2) What is the length of the segment?

3) Difference between segment and vector?

List of sources used:

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

Edited and sent by Poturnak S.A.

We hear the word segment, as a rule, when it comes to geometry or mathematical analysis. In both areas, this word denotes very similar concepts, namely, the part of a straight line that is limited by two points.

Segment in everyday life

Of course, we have to 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 "cut", a person means a piece of this or that material that needs to be cut off 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 straight line bounded by two points, but sometimes you can also find such a term - a set of numbers or points on a straight line between two numbers or points. It sounds much more scientific and complex, but when you think about it, both definitions mean the same thing.

Other meanings

The word "segment" is also pronounced when they want to indicate the passage of a certain stage, for example, "a segment of the path" or "a segment of time." You must have seen such phrases in books.

In addition, the segment after the abolition of serfdom in Russia was called the land plots that the landowners seized from the peasants.

These are the definitions of the word "segment". Learn the meanings of new words in the section.

Straight

The concept of a 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.

Take a ruler and, without lifting your pencil, draw a line of arbitrary length (Fig. 1).

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

Straight lines will be denoted by a small Latin letter, or by two of its points in parentheses (Fig. 2).

The concepts of a 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: It is possible to find at least three points that will not lie on the same line.

Axiom 3: A line always passes through $2$ arbitrary points, and this line is unique.

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

1. The two lines are the same. In this case, each point of one 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 distinct from each other.

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

Line segment

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

Definition 1

A segment will be called a part of a straight line, which is limited by its two arbitrary different points.

Definition 2

The points by which the segment is bounded within the framework of Definition 1 are called the ends of this segment.

The segments will be denoted by its two endpoints in square brackets (Fig. 3).

Segment comparison

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), let's put the end of segment 1 on 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:

Cut length

In addition to comparing segments with others, it is also often necessary to measure segments. To measure a line 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 choosing such a segment, we compare the segments with it, the length of which must be found. Consider an example.

Example 1

Find the length of the next segment

if the next segment is 1

To measure it, we take the segment $$ as a standard. We will postpone it to the segment $$. We get:

Answer: $6$ cm.

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

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

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

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

The easiest way to measure the length of segments is to measure using a ruler.

Example 2

Record the lengths of the following segments:

Let's measure them with a ruler:

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

A point is an abstract object that has no measuring characteristics: no height, no length, no radius. Within the framework of the task, only its location is important

The point is indicated by a number or a capital (large) Latin letter. Several dots - different numbers or different letters so that they can be distinguished

point A, point B, point C

A B C

point 1, point 2, point 3

1 2 3

You can draw three "A" points on a piece of paper and invite the child to draw a line through the two "A" points. But how to understand through which? A A A

A line is a set of points. She only measures length. It has no width or thickness.

Indicated by lowercase (small) Latin letters

line a, line b, line c

a b c

The line could be

1. closed if its beginning and end are at the same point,
2. open if its beginning and end are not connected

closed lines

open lines

You left the apartment, bought bread in the store and returned back to the apartment. What line did you get? That's right, closed. You have returned to the starting point. You left the apartment, bought bread in the store, went into the entrance and talked to your neighbor. What line did you get? Open. You have not returned to the starting point. You left the apartment, bought bread in the store. What line did you get? Open. You have not returned to the starting point.

1. self-intersecting
2. without self-intersections

self-intersecting lines

lines without self-intersections

1. straight
2. broken line
3. crooked

straight lines

broken lines

curved lines

A straight line is a line that does not curve, has neither beginning nor end, it can be extended indefinitely in both directions

Even when a small section of a straight line is visible, it is assumed that it continues indefinitely in both directions.

It is denoted by a lowercase (small) Latin letter. Or two capital (large) Latin letters - points lying on a straight line

straight line a

a

straight line AB

B A

straight lines can be

1. intersecting if they have a common point. Two lines can only intersect at one point.
   • perpendicular if they intersect at a right angle (90°).
2. parallel, if they do not intersect, they do not have a common point.

parallel lines

intersecting lines

perpendicular lines

A ray is a part of a straight line that has a beginning but no end, it can be extended indefinitely in only one direction

The starting point for the beam of light in the picture is the sun.

Sun

The point divides the line into two parts - two rays A A

The beam is indicated by a lowercase (small) Latin letter. Or two capital (large) Latin letters, where the first is the point from which the ray begins, and the second is the point lying on the ray

beam a

a

beam AB

B A

The beams match if

1. located on the same straight line
2. start at one point
3. directed to one side

rays AB and AC coincide

rays CB and CA coincide

C B A

A segment is a part of a straight line that is bounded by two points, that is, it has both a beginning and an end, which means that its length can be measured. The length of a segment is the distance between its start and end points.

Any number of lines can be drawn through one point, including straight lines.

Through two points - unlimited number of curves, but only one straight line

curved lines passing through two points

B A

straight line AB

B A

A piece was “cut off” from the straight line and a segment remained. From the example above, you can see that its length is the shortest distance between two points. ✂ B A ✂

A segment is denoted by two capital (large) Latin letters, where the first is the point from which the segment begins, and the second is the point from which the segment ends

segment AB

B A

Task: where is the line, ray, segment, curve?

A broken line is a line consisting of successively connected segments not at an angle of 180°

A long segment was “broken” into several short ones.

The links of a polyline (similar to the links of a chain) are the segments that make up the polyline. Adjacent links are links in which the end of one link is the beginning of another. Adjacent links should not lie on the same straight line.

The vertices of the polyline (similar to the tops of mountains) are the point from which the polyline begins, the points at which the segments forming the polyline are connected, the point where the polyline ends.

A polyline is denoted by listing all its vertices.

broken line ABCDE

vertex of polyline A, vertex of polyline B, vertex of polyline C, vertex of polyline D, vertex of polyline E

link of broken line AB, link of broken line BC, link of broken line CD, link of broken line DE

link AB and link BC are adjacent

link BC and link CD are adjacent

link CD and link DE are adjacent

A B C D E 64 62 127 52

The length of a polyline is the sum of the lengths of its links: ABCDE = AB + BC + CD + DE = 64 + 62 + 127 + 52 = 305

A task: which broken line is longer, a which one has more peaks? At the first line, all the links are of the same length, namely 13 cm. The second line has all the links of the same length, namely 49 cm. The third line has all the links of the same length, namely 41 cm.

A polygon is a closed polyline

The sides of the polygon (they will help you remember the expressions: "go to all four sides", "run towards the house", "which side of the table will you sit on?") are the links of the broken line. Adjacent sides of a polygon are adjacent links of a broken line.

The vertices of the polygon are the vertices of the polyline. Neighboring vertices are endpoints of one side of the polygon.

A polygon is denoted by listing all its vertices.

closed polyline without self-intersection, ABCDEF

polygon ABCDEF

polygon vertex A, polygon vertex B, polygon vertex C, polygon vertex D, polygon vertex E, polygon vertex F

vertex A and vertex B are adjacent

vertex B and vertex C are adjacent

vertex C and vertex D are adjacent

vertex D and vertex E are adjacent

vertex E and vertex F are adjacent

vertex F and vertex A are adjacent

polygon side AB, polygon side BC, polygon side CD, polygon side DE, polygon side EF

side AB and side BC are adjacent

side BC and side CD are adjacent

side CD and side DE are adjacent

side DE and side EF are adjacent

side EF and side FA are adjacent

A B C D E F 120 60 58 122 98 141

The perimeter of a polygon is the length of the polyline: P = AB + BC + CD + DE + EF + FA = 120 + 60 + 58 + 122 + 98 + 141 = 599

A polygon with three vertices is called a triangle, with four - a quadrilateral, with five - a pentagon, and so on.