In Figure 3 there is a straight line segment. Point, line, straight line, ray, segment, broken line

A straight line is a line (a set of points that have only a length) that is not curved and has neither beginning nor end.

A segment is a straight line bounded at both ends.

The beam is straight and limited at one end.

The point does not have any measuring characteristics; in problems, only its location is important.

Mark three points on the line

A straight line is not a three-dimensional figure; moreover, it does not bend, but continues indefinitely, having neither width nor height in one plane. Therefore, points can be placed anywhere along the entire infinite length; this will only affect the length of the segments cut off by these points.

Number of segments

Since there are three points, we will arrange them arbitrarily on a straight line and call them a, b, c. Thus, three points limit the line, turning them into segments three times, that is, we have three segments

Number of rays

Now let's look at the rays. The straight line is not limited either from the beginning or from the end, but the ray must be limited on one side.

• if we put 1 point on a straight line, respectively, limiting it at this point, we will get 2 rays,
• if we put 2 points, we will limit the line in two places, it would be logical to assume that we will have more than 2 rays, but by limiting it in two places we get a segment, since it is limited on both sides, and 2 rays, since We also have the beginning and end of the line, which are not limited,
• if we put three dots? correct, the situation will repeat itself, only the number of segments will increase

Answer

A straight line on which three points are marked is divided by these points into three segments and two rays.

Let's draw a straight line and mark three points A, B, C on it (see figure)

A segment is a part of a line that consists of all points of this line lying between two given points.

Or simply put, a segment is a part of a line bounded by two points.

The figure shows three segments:

AB (Fig. 1)

AC (Fig. 3)

A ray is a part of a line that consists of all points of this line lying on one side of a given point. Any point on a line divides the line into two rays.

Point A divides the line into rays: a and AC. (Fig. 4)

Point B divides the line into rays: BA and BC. (Fig. 5)

Point C divides the line into rays: CA and c. (Fig. 6)

The result was three segments and six rays.

Line 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, for example, geometric figures and bodies, are formed from the simplest.

If we apply a ruler to two points A and B and draw a line along it connecting these points, we get line segment, which is called AB or VA (we read: “a-be”, “be-a”). Points A and B are called ends of the segment(picture 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 straight 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). Points are designated in capital letters of the 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 figures.

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 = 7cm). In the left column is an indication of the action, in the right column is 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 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;

EQUALS (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; l) 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”.

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

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

point A, point B, point C

point 1, point 2, point 3

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

A line is a set of points. Only the length is measured. It has no width or thickness

Indicated by lowercase (small) Latin letters

line a, line b, line c

The line may 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

1. self-intersecting
2. without self-intersections

self-intersecting lines

lines without self-intersections

1. straight
2. broken
3. crooked

straight lines

broken lines

curved lines

A straight line is a line that is not curved, has neither beginning nor end, it can be continued endlessly in both directions

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

Indicated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters - points lying on a straight line

straight line a

straight line AB

Direct may be

1. intersecting if they have a common point. Two lines can intersect only at one point.
   • perpendicular if they intersect at right angles (90°).
2. Parallel, if they do not intersect, 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 continued indefinitely in only one direction

The ray of light in the picture has its starting point as the sun.

Sun

A point divides a straight line into two parts - two rays A A

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

ray a

beam AB

The rays coincide if

1. located on the same straight line
2. start at one point
3. directed in one direction

rays AB and AC coincide

rays CB and CA coincide

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

Through one point you can draw any number of lines, including straight lines

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

curved lines passing through two points

straight line AB

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 (capital) Latin letters, where the first is the point at which the segment begins, and the second is the point at which the segment ends

segment AB

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

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

A long segment was “broken” into several short ones

The links of a broken line (similar to the links of a chain) are the segments that make up the broken line. 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 a broken line (similar to the tops of mountains) are the point from which the broken line begins, the points at which the segments that form the broken line are connected, and the point at which the broken line ends.

A broken line is designated 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

broken link AB, broken link BC, broken link CD, broken link DE

link AB and link BC are adjacent

link BC and link CD are adjacent

link CD and link DE are adjacent

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

Task: which broken line is longer, A which has more vertices? The first line has all the links of the same length, namely 13 cm. The second line has all links of the same length, namely 49 cm. The third line has all links of the same length, namely 41 cm.

A polygon is a closed polygonal line

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

The vertices of a polygon are the vertices of a broken line. Adjacent vertices are the 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

CD side and DE side are adjacent

side DE and side EF are adjacent

side EF and side FA are adjacent

The perimeter of a polygon is the length of the broken line: 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, etc.

REPEAT THE THEORY

16. Fill in the blanks.

1) A point and a line are examples of geometric shapes.
2) To measure a segment means to count how many single segments fit in it.
3) If you mark point C on segment AB, then the length of segment AB is equal to the sum of the lengths of segments AC + CB
4) Two segments are called equal if they match when superimposed.
5) Equal segments have equal lengths.
6) The distance between points A and B is the length of the segment AB.

SOLVING PROBLEMS

17. Label the segments shown in the figure and measure their lengths.

18. Draw all possible segments with ends at points A, B, C and D. Write down the designations of all the drawn segments.

AB, BC, CD, AD, AC, BD

19. Write down all the segments shown in the figure.

20. Draw segments CK and AD so that CK=4 cm 6 mm, AD=2 cm 5 mm.

21. Draw a segment BE, the length of which is 5 cm 3 mm. Mark point A on it so that BA = 3 cm 8 mm. What is the length of segment AE?

AE = BE-BA = 5 cm 3 mm - 3 cm 8 mm = 1 cm 5 mm

22. Express this value in the indicated units of measurement.

23. Write down the links of the polyline and measure their lengths (in millimeters). Calculate the length of the broken line.

24. Mark point B, located 6 cells to the left and 1 cell below point A; point C, located 3 cells to the right and 3 cells below point B; point D, located 7 cells to the right and 2 cells above point C. Connect points A, B, C and D in series with segments.

A broken ABCD was formed, consisting of 3 links.

25. Calculate the length of the broken line shown in the figure.

a) 5*36 = 180 mm
b) 3*28 = 84 mm
c) 10*10+15*4 = 160 mm

26. Construct a broken line DCEC so that DC=18 mm, CE=37 mm, EK=26 mm. Calculate the length of the broken line.

27. It is known that AC = 17 cm, ВD = 9 cm, ВС = 3 cm. Calculate the length of the segment AD.

28. It is known that MK=KN=NP=PR=RT=3 cm. What other equal segments are there in this figure? Find their lengths.

29. Mark points on a straight line so that the distance between any two neighboring points is 4 cm, and between the extreme points is 36 cm. How many points are marked?

30. Draw, without lifting the pencil from the paper, the figures shown in the figure. Each line can be drawn with a pencil only once.