What is ray drawing. link BC and link CD are adjacent

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

Point in mathematics

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

In the figure you can see an 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 the set of all points lying on a straight line between the ends of the 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.

Direct in mathematics

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

The figure shows two straight lines: CD and EF.

Beam 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 starting point of the beam, so letters cannot be swapped.

The figure shows the rays: DC, KC, EF, MT, MS. Beams KC and KD are 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 the number line, as well as the OD and ED rays

We all once studied geometry at school, but not all of us remember what a segment is. And even more so, few people can explain the concept of rays and how they are designated. Let's try in this article to remind ourselves of these definitions and consider them in mathematics. We will also define what a beam is and how it differs from light. If you get into it, it won't be difficult to understand.

Definition of concepts

First, let's remember what is called geometry. Geometry is a branch of mathematics that studies geometric figures and their properties. These include a triangle, square, rectangle, parallelepiped, circle, oval, rhombus, cylinder, etc. The simplest figure is a straight line. It is endless and has no beginning. Two lines will intersect only at one single point. Countless straight lines can be drawn through one point. Every point on a line divides it into two.

It consists of points located on one side. All concepts of these subsets can be named this way. The ray is denoted by one lowercase Latin letter or two capital letters, when one point is the beginning (for example, O), and the second lies on it (for example, F, K and E).

A geometric figure with angles is based on half-lines. They start at the point where they intersect, but the other side is directed to infinity. The beginning divides the line into 2 parts. In writing it is usually referred to as two capitals (OF) or one Latin letter (a, b, c). If a straight line is given, then OB is written in rounded brackets: (OB). If this is a segment - in square brackets.

Thus, a ray is part of a straight line. Through any point you can draw many straight lines, but through 2 non-coinciding ones - only one. The latter can interact only in three ways: intersect, cross, or be parallel to each other. There are linear equations that define a straight line on a plane.

Notation in geometry

There are several designation options:

Need to know: What is horizontal position?

The difference between light rays and geometric ones

In geometry, these concepts are very similar. A ray is a line, but it is the energy of light. In other words, it is a small beam of light. In optics, this concept, like the concept of a straight line, is basic in geometry. The light does not have a concentrated direction, diffraction occurs. But when the light flux is very strong, divergence is neglected and a clear direction can be identified.

the beginning of the ray.

a ABOUT

ray k.

semi-straight.

Task:

The figure shows that these conditions are met by rays AB and AC, as well as rays BC and BA. Therefore, they are coincident.

Answer: AB and AC, BC and BA.

Along with such concepts as point, segment, line, there is one more concept in geometry. It is called ray. A ray is a part of a straight line, limited on one side by a point, and on the other side - infinite, i.e. not limited by anything.

An analogy can be drawn with nature. For example, a beam of light that we can direct 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 at which it begins. It is called the beginning of the ray.

If we take an arbitrary straight line a, and mark some point on it ABOUT, then this point will split our line into two parts. Each of which will be a ray. 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 designated by one Latin letter. The figure below shows ray k.

You can also denote the 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 the 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 indicate the starting point of the ray and the line to which this ray belongs. For example, the figure below shows the ray Ok.

Sometimes they say that the ray comes from point O. This means that point O is the beginning of the ray. Rays are also sometimes called semi-straight.

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 coinciding rays.

The 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 these conditions are met by rays AB and AC, as well as rays BC and BA. Therefore, they are coincident.

From the school geometry course, few people have accurate information about what a segment is, how it is designated, what a broken line, a straight line, a point are, and how rays are designated. If you cannot 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 figures and their properties. There is a lot of information, sometimes there is not enough time to take in and remember everything. Some knowledge needs to be refreshed after several 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, limited on one side by a point, and on the other hand free, that is, without restrictions. To quickly remember how rays are designated and what they look like, you can give a simple example: we can direct a beam of light from a flashlight into the sky, right? On one side, the beam is limited - from the place where it comes out, that is, from the flashlight. On the other hand, it has no restrictions. It turns out that there is only one extreme point of the beginning of the ray, it is called the “beginning”. The second point does not exist, because the beam goes to infinity.

To understand how to mark 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 will put a limit - a dot, this is the beginning of the ray. There will be no second point at the end of the segment.

How are the rays designated?

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

There are several designation options:

• Let's draw a straight line in a notebook and mark the point of origin of the ray. And let's give it a name. For example, let it be beam "C". The first point is the beginning of the ray; the second point, as you already remembered, does not exist. This is the classic ray notation scheme.
• The second option is more interesting: the beam can be designated 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. Let’s say that on a segment 10 cm long, the beginning of the ray is designated by the letter A, and at a distance of 4 cm from the beginning of the ray there is a second point, point B. Then the ray should be designated as ray “AB”. To make it clearer, you can read it like this: the second point B is the point through which the ray passes.
• The rays can also be designated in a third way, when the starting point is not at the beginning of the ray, but with a slight deviation. For example, draw a straight line 10 cm long, step back 1 cm from the left edge, put a dot - this will be the beginning of the ray. We denote, for example, the letter O. We do not put a point in the middle of the ray, but we denote this part of the ray with the letter K. In this case, the letter O will be the beginning of this ray, it comes from this point. The beam is read like this: “OK”, it is semi-direct.

How is a beam indicated in a notebook?

The designation on the letter of the ray must be remembered once: the rays are written in Latin capital letters. If it is a straight line, then you need to write the ray AB in parentheses: (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 often happens, the word has quite a few meanings. It’s worth taking a closer look at the most key ones.

Geometry

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

It is quite difficult to define this term, since it is one of the original 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 matter. However, a straight line can be interpreted as a line between two points.

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

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

So, it's worth returning to the concept of a ray. It is part of a straight line. If you put a point on such a line, you will automatically get two rays, and they will not have a second point limiting them.

Thus, ray is part of a straight line having a beginning but no end.

Light beam

Geometric optics treats the concept of a light beam in a fairly 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.

Just 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, a 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 basic terms in the exact sciences, this word refers to a wide variety of objects. For example, about seven sports clubs had this name, and some of them still exist. Many villages, towns and hamlets in Russia, Ukraine and Belarus are also called Luchi. Ships are not far 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-seaters and are used for racing. They are often used as educational equipment for children, but competitions are also held on them.

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

A B C

point 1, point 2, point 3

1 2 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

a b 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

You left the apartment, bought bread at the store and returned back to the apartment. What line did you get? That's right, closed. You are back to your starting point. You left the apartment, bought bread at the store, went into the entrance and started talking with your neighbor. What line did you get? Open. You haven't returned to your starting point. You left the apartment and bought bread at the store. What line did you get? Open. You haven't returned to your starting point.

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

a

straight line AB

B A

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

a

beam AB

B A

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

C B A

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

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 (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

B A

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

A B C D E 64 62 127 52

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 polyline

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

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

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.

In dealing with the personality horoscope of the average non-aspirant, the astrologer must try to ascertain his Ray of personality by examining his temperament, physical characteristics, emotional characteristics, type of mind and environment. This will allow him to create a much more useful chart with the orthodox planets that rule life. In the case of the disciple's horoscope, he should do the same by trying to establish the Soul Ray. This Ray manifests its qualities and essence only in advanced people, and when it is obvious that a person is clearly a disciple, then the esoteric planets rule his chart. Having determined the Ray of a person undergoing tests in Scorpio, the astrologer can connect other Rays with him and with his experience

We have considered the Rays that directly affect our planet, are focused through the three governing planets and emanate from certain constellations. Ultimately, the planet is the result, or consequence, (perhaps it is better to say, the final consequence) of ray influences, just as in man the physical body is also a consequence of the governing Rays. Certain powers manifest themselves through the planets. There are only three of them, the so-called sacred planets - these are those ray forces that express the soul and spirit, and the Ray of personality of the great saturating Life, the Planetary Logos, is subordinate to two higher Rays, just as in the case of a person who has passed the third initiation. A non-sacred planet such as Earth is temporarily subordinate to the personality Ray of saturating Life, and therefore the esoteric monadic Ray is not yet effective in this case.

It should be mentioned that the two main divisions of humanity, the West and the East, are also governed by certain ray energies:

West Soul Ray 2nd Ray

Personality Ray 4th Ray

East Soul Ray 4th Ray

Personality Ray 3rd Ray

The West and the East are united through the Personality Ray of the West and the Egoic Ray of the East, indicating the ultimate understanding that will be achieved once the Second Soul Ray of the West becomes the dominant factor. When these various relationships have been sufficiently understood by the peoples of the world, you will have the key to many of the events taking place today and will understand more clearly the purpose and method of the ongoing process.

So-called Sun sign indicates the physical, mental and spiritual nature of a person. It contains the secret of the Ray of personality and a human response or lack of response to one's soul, or a real person. It also indicates the integration already achieved and the point of revelation of the qualities of the soul, the existing equipment, quality of life and directly possible group relations.

Over the course of centuries, man passes from sign to sign, while a particular sign is determined by the nature of its personality ray, which, as you know, itself changes from incarnation to incarnation.

Do not forget that the Rays of personality change from period to period, both in relation to individuals and in relation to countries and cities:

RAYS

City Soul Personality Sign

1. London 5th 7th Gemini

2. New York 2nd 3rd Cancer

3. Tokyo 6th 4th Cancer

4. Geneva 1st 2nd Leo

5. Darjeeling 2nd 5th Scorpio

Table. Three lists of planetary rulers

	Constellation	Orthodox	For the student	For Creative Hierarchies
		Mars	Mercury
	Twins	Mercury		Earth
		Moon
		Sun	Sun	Sun
		Mercury	Moon
	Scorpion	Mars	Mars	Mercury
			Earth	Mars
				Moon
			Pluto	Pluto

In time it will be confirmed that in the life cycle of humanity through which we are now passing, the centers are governed by the following rays and therefore by the planets:

AVERAGE HUMANITY - EXOTERIC PLANETS

1. Head center 1st Ray Pluto

2. Ajna Center 5th Ray Venus

3. Throat center 3rd Ray Earth

4. Heart center 2nd Ray Sun

5. Solar Plexus Center 6th Ray Mars

6. Sacral center 7th Ray Uranus

7. Base of the spine 1st Ray Pluto

DISCIPLES INSTITUTED - ESOTERIC PLANETS

1. Sahasrara 1st Ray Vulcan

2. Ajna 5th Ray Venus

3. Vishuddha 3rd Ray Saturn

4. Anahata 2nd Ray Jupiter

5. Manipura 6th Ray Neptune

6. Svadhisthana 7th Ray Uranus

7. Muladhara 1st Ray Pluto