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

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

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.

Point and line are the main geometric figures on the plane.

The ancient Greek scientist Euclid said: “a point” is that which has no parts.” The word "point" in Latin means the result of an instant touch, a prick. The point is the basis for constructing any geometric figure.

A straight line or just a straight line is a line along which the distance between two points is the shortest. A straight line is infinite, and it is impossible to depict the entire line and measure it.

Points are denoted by capital Latin letters A, B, C, D, E, etc., and straight lines by the same letters, but lowercase a, b, c, d, e, etc. A straight line can also be denoted by two letters corresponding to points lying on her. For example, the line a can be denoted by AB.

We can say that the points AB lie on the line a or belong to the line a. And we can say that the line a passes through the points A and B.

The simplest geometric figures on a plane are a segment, a ray, a broken line.

A segment is a part of a line, which consists of all points of this line, bounded by two selected points. These points are the ends of the segment. A segment is indicated by indicating its ends.

A ray or half-line is a part of a line, which consists of all points of this line, lying on one side of its given point. This point is called the starting point of the half-line or the beginning of the ray. A ray has a start point but no end point.

Half-lines or rays are denoted by two lowercase Latin letters: the initial and any other letter corresponding to a point belonging to the half-line. In this case, the starting point is placed in the first place.

It turns out that the line is infinite: it has neither beginning nor end; a ray has only a beginning but no end, while a segment has a beginning and an end. Therefore, we can only measure a segment.

Several segments that are connected in series with each other so that the segments (adjacent) having one common point are not located on the same straight line represent a broken line.

The polyline can be closed or open. If the end of the last segment coincides with the beginning of the first, we have a closed broken line, if not, an open one.

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the beginning of the beam.

a O

beam k.

semi-direct.

A task:

The figure shows that the beams AB and AC, as well as the beams BC and BA, satisfy these conditions. Therefore, they are matched.

Answer: AB and AC, BC and BA.

Along with such concepts as a point, a segment, a line, there is another concept in geometry. It is called beam. A ray is a part of a straight line, limited on one side by a point, and on the other side - infinite, i.e. nothing limited.

You can draw an analogy with nature. For example, a beam of light that we can send from earth into space. On the one hand, it is limited, but on the other hand, it is not. Each ray has one extreme point where it starts. It is called the beginning of the beam.

If we take an arbitrary line a, and mark some point on it O, then this point will divide our line into two parts. Each of which will be a beam. Point O will belong to each of these rays. Point O will be in this case the beginning of these two rays.

The beam is usually denoted by one Latin letter. The figure below shows beam k.

It is also possible to designate a beam with two capital Latin letters. In this case, the first of them is the point at which the beginning of the beam lies. The second is a point that belongs to the ray, or in other words - through which the ray passes.

The figure shows the OS beam.

Another way to designate a ray is to specify the starting point of the ray and the line to which the ray belongs. For example, the figure below shows the beam Ok.

It is sometimes said that the ray comes from the point O. This means that the point O is the beginning of the ray. Rays are sometimes also called semi-direct.

A task:

Draw a straight line and mark points A B on it and mark point C on segment AB. Among the rays AB, BC, CA, AC and BA, find pairs of matching rays.

Rays coincide if they lie on the same straight line and have a common origin, and none of them is a continuation of another ray.
The figure shows that the beams AB and AC, as well as the beams BC and BA, satisfy these conditions. Therefore, they are matched.

From the school geometry course, few people have accurate information about what a segment is, how it is denoted, what a broken line, a straight line, a point are, and how rays are denoted. If you can not remember the initial geometry course, just read this article.

What is geometry? This is a mathematical section in which the student gets acquainted with geometric shapes and their properties. There is a lot of information, sometimes there is not enough time to cover and remember everything. Some knowledge needs to be refreshed after a few months and even years. For example, remember what rays are and how they are designated.

What is a ray in geometry

A ray is a straight line, on one side limited by a point, and on the other side - free, that is, without restrictions. To quickly remember how the rays are designated and what they look like, we can give a simple example: can we send a ray of light from a flashlight into the sky? On the one hand, the beam is limited - from the place where it comes from, that is - from the flashlight. On the other hand, it has no limits. It turns out that there is only one extreme point of the beginning of the beam, and it is called the “beginning”. The second point does not exist because the ray goes to infinity.

To understand how to designate a ray on a piece of paper, you need to draw a straight line. For example, let it be a segment equal to 10 cm. On the right side, we put a restriction - a point, this is the beginning of the beam. There will be no second point at the end of the segment.

How are rays defined?

Let's continue to remember what a beam is and how to designate it.

There are several notation options:

• Let's draw a straight line in a notebook, denote the point of the beginning of the beam. And give her a name. For example, let it be ray "C". The first point is the beginning of the beam, the second point, as you already remembered, does not exist. This is a classic ray designation scheme.
• The second option is more interesting: the beam can be denoted by several letters. For example, there can be 2 letters on one beam. The first is the beginning of the beam, let it be the letter A, and the second can be located with a certain step. Suppose, on a segment 10 cm long, the beginning of the beam is marked with the letter A, and at a distance of 4 cm from the beginning of the beam there is a second point, point B. Then the beam must be designated as the beam "AB". To make it clearer, you can read this: the second point B is the point through which the beam passes.
• Rays can also be designated in a third way, when the starting point will not be at the beginning of the ray, but with a slight deviation. For example, we draw a straight line 10 cm long, retreat from the left edge 1 cm, put a point - this will be the beginning of the beam. We denote, for example, the letter O. We do not put a point in the middle of the beam, but we denote this part of the beam with the letter K. In this case, the letter O will be the beginning of this beam, it comes from this point. The beam is read like this: "OK", it is half-line.

How is a beam indicated in a notebook

The designation on the letter of the beam must be remembered once: the rays are written in Latin capital letters. If it is a straight line, then you need to write the beam AB in round brackets: (AB). If you have a segment in front of you, then it is written only in square brackets.

Most often, this question is asked in schools, in geometry lessons, and the concept is also quite popular in optics. However, as is often the case, the word has quite a few meanings. It is worth dwelling in more detail on the most key ones.

Geometry

In order to understand what a ray is from the point of view of geometry, it is necessary to consider one of the fundamental concepts of this science, namely, a straight line.

It is rather difficult to define this term, since it is one of the initial ones, and it is with the help of a straight line that other various words are explained. There are quite a few axioms on this subject. However, a straight line can be interpreted as a line between two points.

The straight line has its own properties, according to Euclidean geometry.

• Through any point, you can draw as many lines as you like, but through two non-coinciding points - only one.
• Lines can only be in three states - they can intersect, be parallel to each other, and they can also intersect.
• There is a linear equation defining a straight line on a plane.

So, it is worth returning to the concept of a ray. It is part of a straight line. If a point is placed on such a line, then two rays will automatically be obtained, while they will not have a second point limiting them.

In this way, a ray is part of a line having a beginning but no end.

light beam

Geometric optics treats the concept of a light beam in a rather similar way. Here it will also be a line, but it will be used by light energy. In other words, a light beam is small beam of light.

Like the concept of a straight line in geometry, the concept of a ray in optics is a fairly basic phenomenon. However, unlike a geometric beam, the light beam does not have any clear direction, since diffraction occurs. However, if the light is very large, then the divergence is usually neglected. In this case, a clear direction can be identified.

In addition to the basic terms in the exact sciences, this word denotes a wide variety of objects. For example, about seven sports clubs had this name, and some of them still exist. Many villages, towns and farms in Russia, Ukraine and Belarus are also called Rays. Vessels do not lag behind them - and in this case Luch is a brand of passenger ships, as well as a whole class of yachts.

These yachts are single and used for racing. Often they are used as a training projectile for children, but competitions are also held on it.

Along with such concepts as a point, a segment, a line, there is another concept in geometry. It is called beam. A ray is a part of a straight line, limited on one side by a point, and on the other side - infinite, i.e. nothing limited.

You can draw an analogy with nature. For example, a beam of light that we can send from earth into space. On the one hand, it is limited, but on the other hand, it is not. Each ray has one extreme point where it starts. It is called the beginning of the beam.

If we take an arbitrary line a, and mark some point on it O, then this point will divide our line into two parts. Each of which will be a beam. Point O will belong to each of these rays. Point O will be in this case the beginning of these two rays.

The beam is usually denoted by one Latin letter. The figure below shows beam k.

It is also possible to designate a beam with two capital Latin letters. In this case, the first of them is the point at which the beginning of the beam lies. The second is a point that belongs to the ray, or in other words - through which the ray passes.

The figure shows the OS beam.

Another way to designate a ray is to specify the starting point of the ray and the line to which the ray belongs. For example, the figure below shows the beam Ok.

It is sometimes said that the ray comes from the point O. This means that the point O is the beginning of the ray. Rays are sometimes also called semi-direct.

A task:

Draw a straight line and mark points A B on it and mark point C on segment AB. Among the rays AB, BC, CA, AC and BA, find pairs of matching rays.

Rays coincide if they lie on the same straight line and have a common origin, and none of them is a continuation of another ray.
The figure shows that the beams AB and AC, as well as the beams BC and BA, satisfy these conditions. Therefore, they are matched.