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

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.

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

A segment is a straight line bounded at both ends.

A beam is a straight line limited at one end.

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

Mark three points on the line

A straight line is not a three-dimensional figure, moreover, it does not curve, but continues indefinitely, having neither width nor height in 1 plane. Therefore, points can be placed throughout the entire infinite length anywhere, this will only affect the length of the segments cut off by these points.

Number of segments

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

Number of beams

Now let's deal with the rays. The straight line is not limited from the beginning or from the end, and the ray must be limited on one side.

• if we put 1 point on a straight line, respectively limiting it at this point, we 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 limiting in two places we got a segment, since it is limited on both sides, and 2 rays, because we also have the beginning and end of the straight line, which are not limited,
• if we put three dots? right, the situation will repeat itself, only the number of segments will increase

Answer

The 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 on it three points A, B, C. (see figure)

A segment is a part of a straight line that consists of all points of this straight line that lie between two given points on it.

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

The figure has three segments:

AB (Fig. 1)

AC (Fig. 3)

A ray is a part of a line that consists of all points of this line that lie on the same 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)

It turned out three segments and six rays.

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, ADorDA,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".

REPEATING THE THEORY

16. Fill in the blanks.

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

SOLVING PROBLEMS

17. Mark 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 segments drawn.

AB, BC, CD, AD, AC, BD

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

20. Draw segments SK and AD so that SK=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= 5cm 3mm - 3cm 8mm = 1cm 5mm

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 polyline.

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 line ABCD was formed, consisting of 3 links.

25. Calculate the length of the polyline 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, EC=26 mm. Calculate the length of the polyline.

27. It is known that AC=17 cm, BD=9 cm, BC=3 cm. Calculate the length of 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. Points were marked on a straight line so that the distance between any two neighboring points is 4 cm, and between the extreme points - 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.