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 tops 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.

Hello, dear readers of the blog site. One of the concepts of geometry, which is introduced in elementary school, is a line segment. A lot of problems in mathematics and geometry are based on the concepts of a segment and a straight line.

Understanding what a segment is will help you solve all kinds of problems and examples in mathematics lessons both at school and in higher educational institutions.

A segment is a geometric figure

According to the definition in the dictionary, a segment is called part of the line bounded by two points on it. It is by designation of these points that the name of the segment is given.

The figure below shows segment AB. Points A and B are the ends of the segment. The length of a segment is the distance between its ends.

In mathematics, it is customary to designate points, and, accordingly, segments, with capital letters of the Latin alphabet. If you need to draw a segment, most often it is depicted without a straight line, but only from one end to the other.

You can also say that the segment - is the collection of all points, which lie on one straight line and are located between two given points, which are the ends of the given segment.

If one more point is marked on the segment between its ends, it will divide this segment into two. The length of segment AB can be calculated by summing the lengths of segments AC and CB.

Difference between line segment, ray and line

Schoolchildren sometimes confuse the concepts of a straight line, a ray and a segment. Indeed, these concepts are very similar to each other, but they have a fundamental difference:

1. Straight called a line that does not curve, and also has no beginning and end.
2. Ray is the part of a straight line bounded by one point. It has a beginning and has no end.
3. limited to two points. It has both a beginning and an end.

A point on a line divides it into two rays. The number of segments on one straight line can be infinite.

To distinguish between these figures in the figure, dots are put or not put at the beginning and end of the drawn line. When drawing a ray, a point is placed at one end, and when depicting a segment, at both ends. The line has no ends, so no points are placed at the end of the line.

Directed segment is a vector

Segments are of two types:

1. Non-directional.
2. Directed.

For non-directional segments, AB and BA are the same segments, since the direction does not matter.

If we talk about directed segments, the order of listing its ends is crucial. In this case, AB ➜ and BA ➜ are different segments, since they are oppositely directed.

Directional segments are called vectors. Vectors can be denoted either by two capital letters of the Latin alphabet with an arrow above them, or by one small letter with an arrow.

The module of a vector is the length of a directed segment. Designated as AB ➜ . Modules of vectors AB ➜ and BA ➜ are equal.

Vectors are often considered in a coordinate system. The modulus of a vector is equal to the square root of the sum of the squares of the coordinates of the ends of the vector.

Collinear vectors are those that lie on one or parallel lines.

A broken line is a set of connected segments

A broken line consists of many segments, which are called its links. These segments are connected to each other at their ends and are not located at an angle of 180°.

The vertices of the polyline are the following points:

1. The point from which the polyline began.
2. The point at which the polyline ended.
3. Points at which adjacent links are connected (polyline segments).

The number of vertices of a polyline is always one more than the number of its links. A polyline is denoted by listing all its vertices, starting from one end and ending at the other.

For example, the polyline ABCDEF consists of the segments AB, BC, CD, DE and EF and the vertices A, B, C, D, E and F. The links AB and BC are adjacent because they have a common end - point B. The length of the polyline is calculated as the sum of the lengths of all its links.

Any closed polyline is a geometric figure - a polygon.

The sum of the angles of a polygon is a multiple of 180° and is calculated using the following formula 180*(n-2), where n is the number of angles or segments that make up the figure.

Time interval

Interestingly, the word segment is applicable not only to geometric concepts, but also as a temporary term.

A period of time is the period between two events, dates. It can be measured in seconds or minutes, or years or even decades.

Time as a whole in this case is defined as a time line.

Good luck to you! See you soon on the blog pages site

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

Section

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.

Section 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

Section- 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. Mathematics. 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.

We will look at each of the topics, and at the end there will be tests on the topics.

Point in math

What is a point in mathematics? A mathematical point has no dimensions and is indicated by capital Latin letters: A, B, C, D, F, etc.

In the figure, you can see the image of points A, B, C, D, F, E, M, T, S.

Segment in mathematics

What is a segment in mathematics? In mathematics lessons, you can hear the following explanation: a mathematical segment has a length and ends. A segment in mathematics is a set of all points lying on a straight line between the ends of a segment. The ends of the segment are two boundary points.

In the figure we see the following: segments ,,,, and , as well as two points B and S.

Straight lines in mathematics

What is a straight line in mathematics? Definition of a straight line in mathematics: a straight line has no ends and can continue in both directions to infinity. A straight line in mathematics is denoted by any two points on a straight line. To explain the concept of a straight line to a student, we can say that a straight line is a segment that does not have two ends.

The figure shows two straight lines: CD and EF.

Ray in mathematics

What is a ray? Definition of a ray in mathematics: A ray is a part of a line that has a beginning and no end. The name of the beam contains two letters, for example, DC. Moreover, the first letter always indicates the point of the beginning of the beam, so you cannot swap the letters.

The figure shows the beams: DC, KC, EF, MT, MS. Beams KC and KD - one beam, because they have a common origin.

Number line in mathematics

Definition of a number line in mathematics: A line whose points mark numbers is called a number line.

The figure shows a number line, as well as a ray OD and ED