What is a ray and how are rays designated? There are other meanings

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's called 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, and 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 ray 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.

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.

This is interesting: how area is designated, examples for calculation.

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 and 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.

Goals:

Introduce students to the concept of a ray as an infinite figure;
Learn to show the beam using a pointer;
Continue building computing skills;
Improve problem solving skills;
Develop the ability to analyze and generalize.

Lesson progress

I. Organizational moment.

Guys, are you ready for the lesson? ( Yes. )
I count on you, friends!
You are a good friendly class.
Everything will work out for you!

II. Motivation for learning activities.

I really want the lesson to be interesting, informative, so that together we repeat and consolidate what we already know and try to discover something new.

III.Updating knowledge.

Read the numbers and name the “extra” number in each row:
a) 90, 30, 40, 51.60;
b) 88, 64,55,11, 77, 33;
c) 47, 27, 87, 74, 97, 17;
List the numbers in order:
a) from 20 to 30;
b) from 46 to 57;
c) from 75 to 84;
Do you think these texts will be tasks?

Change the question in the second text so that it becomes a task.

Change the condition so that the text becomes a task.

Solve the given problems.

IV. Primary assimilation of new knowledge.

Draw a line like this.

What is it called?

Draw a line like this.

What is it called? What is the difference between a segment and a straight line?

Draw a line like this.

Who knows what it's called?

Look at the picture, you see similar lines, what is it?

This line is called a ray. How does it differ from a straight line and a segment?

This is a very interesting figure: it has a beginning and no end.

And this is how they portray her. ( Work on the board and in notebooks.) Mark a point, apply a ruler to it and draw a line along the ruler.

No matter how long the ruler is, we still won’t be able to draw the entire beam. In the figure we have depicted only part of the beam, which shows the direction of the beam.

The beam can be drawn in any direction:

Draw three different rays in your notebook.

To distinguish one ray from another, we will agree to denote the ray with two letters of the Latin alphabet in the same way as we denoted segments. The letters must be written in a strictly defined order: the first letter is written that indicates the beginning of the beam, the second is written above or below the beam.

Look at the picture in the textbook. The red beam is indicated by two letters. What letter indicates the beginning of the ray?

Let's read the entry together: “Beam AB”

Now read the following entries: beam BC, beam MK, beam BA, beam OX.

It is important to learn how to show the beam correctly. We will do this with the end of the pointer. ( Demonstration by the teacher.)

Now look at the poster. ( Prepared in advance, it has 3 rays.) It shows 3 rays. Read the title of each one. When naming a beam, show it with a pointer.

Fizminutka

1, 2, 3, 4, 5
We all know how to count.
We also know how to relax:
Let's put our hands behind our backs,
Let's raise our heads higher
And let's breathe easily.
One, two - head up,
Three, four - the legs are wider,
Five, six - quiet network.
Once - get up, stretch.
Two – bend over, straighten up.
Three - three claps of your hands,
Three nods of the head.
By four – your arms are wider.
Five - wave your arms.
Six - sit quietly at your desk.

V.Initial check of understanding.

1) Working with the textbook.

Is it possible to draw the entire beam?

In what direction can the ray be drawn?

Students name each ray by first reading the letter corresponding to the beginning of the ray.

Students draw a ray in their notebooks and label it with letters.

Place point O in your notebook. Draw a straight line through it. How many rays did you get?

Draw another straight line through this point. How many rays are there now?

VI. Organization of mastering methods of activity.

1) Work in a printed notebook.

Differentiated task.

1st group - No. 19

2nd group - No. 20

3rd group - No. 21

2) Fizminutka – ophthalmic simulator.

3) Working from the textbook

Read what addition methods did Znayka come up with?

Find the results of addition using the same methods.

What is known about the problem?

What do you need to know?

In short – is it more or less?

How to find out the length of a pencil?

Write down your answer.

VII. Reflection.

What new did you learn in the lesson?

What is a beam?

How to draw a ray?

How many rays can be drawn through one point?

Today in class they helped me.....

VIII. Homework.

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

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.

Solution:

Answer: AB and AC, BC and BA.

A point and a straight line are the basic geometric figures on a plane.

The ancient Greek scientist Euclid said: “a point” is something that has no parts.” The word “point” translated from Latin means the result of an instant touch, an injection. A point is the basis for constructing any geometric figure.

A straight line or simply 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 straight 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, straight line a can be designated AB.

We can say that points AB lie on line a or belong to line a. And we can say that straight line a passes through 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 that consists of all points of this line, limited 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 that consists of all points of this line lying on one side of a given point. This point is called the starting point of the half-line or the beginning of the ray. The beam has a starting point, but no end.

Half-lines or rays are designated 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 straight line is infinite: it has neither beginning nor end; a ray has only a beginning, but no end, but a segment has a beginning and an end. Therefore, we can only measure a segment.

Several segments that are sequentially connected to each other so that the segments (neighboring) that have one common point are not located on the same straight line represent a broken line.

A broken line 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, it is an open line.

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