Axial symmetry drawings on paper. Central symmetry

Scientific and practical conference

Municipal educational institution "Secondary" secondary school No. 23"

city ​​of Vologda

section: natural science

design and research work

TYPES OF SYMMETRY

The work was completed by an 8th grade student

Kreneva Margarita

Head: higher mathematics teacher

2014

Project structure:

1. Introduction.

2. Goals and objectives of the project.

3. Types of symmetry:

3.1. Central symmetry;

3.2. Axial symmetry;

3.3. Mirror symmetry (symmetry about a plane);

3.4. Rotational symmetry;

3.5. Portable symmetry.

4. Conclusions.

Symmetry is the idea through which man has tried for centuries to comprehend and create order, beauty and perfection.

G. Weil

Introduction.

The topic of my work was chosen after studying the section “Axial and central symmetry” in the course “8th grade Geometry”. I was very interested in this topic. I wanted to know: what types of symmetry exist, how they differ from each other, what are the principles for constructing symmetrical figures in each type.

Purpose of the work : Introduction to different types of symmetry.

Tasks:

Study the literature on this issue.

Summarize and systematize the studied material.

Prepare a presentation.

In ancient times, the word “SYMMETRY” was used to mean “harmony”, “beauty”. Translated from Greek, this word means “proportionality, proportionality, sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.

There are two groups of symmetries.

The first group includes symmetry of positions, shapes, structures. This is the symmetry that can be directly seen. It can be called geometric symmetry.

The second group characterizes symmetry physical phenomena and the laws of nature. This symmetry lies at the very basis of the natural scientific picture of the world: it can be called physical symmetry.

I'll stop studyinggeometric symmetry .

In turn, there are also several types of geometric symmetry: central, axial, mirror (symmetry relative to the plane), radial (or rotary), portable and others. Today I will look at 5 types of symmetry.

Central symmetry

Two points A and A 1 are called symmetrical with respect to point O if they lie on a straight line passing through point O and are located along different sides at the same distance from it. Point O is called the center of symmetry.

The figure is said to be symmetrical about the pointABOUT , if for each point of the figure there is a point symmetrical to it relative to the pointABOUT also belongs to this figure. DotABOUT called the center of symmetry of a figure, the figure is said to have central symmetry.

Examples of figures with central symmetry are a circle and a parallelogram.

The figures shown on the slide are symmetrical relative to a certain point

2. Axial symmetry

Two pointsX And Y are called symmetrical about a straight linet , if this line passes through the middle of the segment XY and is perpendicular to it. It should also be said that each point is a straight linet is considered symmetrical to itself.

Straightt – axis of symmetry.

The figure is said to be symmetrical about a straight linet, if for each point of the figure there is a point symmetrical to it relative to the straight linet also belongs to this figure.

Straighttcalled the axis of symmetry of a figure, the figure is said to have axial symmetry.

An undeveloped angle, isosceles and equilateral triangles, a rectangle and a rhombus have axial symmetry.letters (see presentation).

Mirror symmetry (symmetry about a plane)

Two points P 1 And P are called symmetrical relative to the plane a if they lie on a straight line perpendicular to the plane a and are at the same distance from it

Mirror symmetry well known to every person. It connects any object and its reflection in a flat mirror. They say that one figure is mirror symmetrical to another.

On a plane, a figure with countless axes of symmetry was a circle. In space, a ball has countless planes of symmetry.

But if a circle is one of a kind, then in the three-dimensional world there is a whole series of bodies with an infinite number of planes of symmetry: a straight cylinder with a circle at the base, a cone with a circular base, a ball.

It is easy to establish that every symmetrical plane figure can be aligned with itself using a mirror. It is surprising that such complex figures as a five-pointed star or an equilateral pentagon are also symmetrical. As this follows from the number of axes, they are distinguished by high symmetry. And vice versa: it is not so easy to understand why such a seemingly regular figure, like an oblique parallelogram, is asymmetrical.

4. P rotational symmetry (or radial symmetry)

Rotational symmetry - this is symmetry, the preservation of the shape of an objectwhen rotating around a certain axis through an angle equal to 360°/n(or a multiple of this value), wheren= 2, 3, 4, … The indicated axis is called the rotary axisn-th order.

Atn=2 all points of the figure are rotated through an angle of 180 0 ( 360 0 /2 = 180 0 ) around the axis, while the shape of the figure is preserved, i.e. each point of the figure goes to a point of the same figure (the figure transforms into itself). The axis is called the second-order axis.

Figure 2 shows a third-order axis, Figure 3 - 4th order, Figure 4 - 5th order.

An object can have more than one rotation axis: Fig. 1 - 3 axes of rotation, Fig. 2 - 4 axes, Fig. 3 - 5 axes, Fig. 4 – only 1 axis

Everyone famous letters“I” and “F” have rotational symmetry. If you rotate the letter “I” 180° around an axis perpendicular to the plane of the letter and passing through its center, the letter will align with itself. In other words, the letter “I” is symmetrical with respect to a rotation of 180°, 180°= 360°: 2,n=2, which means it has second-order symmetry.

Note that the letter “F” also has second-order rotational symmetry.

In addition, the letter has a center of symmetry, and the letter F has an axis of symmetry

Let's return to examples from life: a glass, a cone-shaped pound cake with ice cream, a piece of wire, a pipe.

If we take a closer look at these bodies, we will notice that all of them, in one way or another, consist of a circle, through an infinite number of symmetry axes there are countless symmetry planes. Most of these bodies (they are called bodies of rotation) also have, of course, a center of symmetry (the center of a circle), through which at least one rotational axis of symmetry passes.

For example, the axis of the ice cream cone is clearly visible. It runs from the middle of the circle (sticking out of the ice cream!) to the sharp end of the funnel cone. We perceive the totality of symmetry elements of a body as a kind of symmetry measure. The ball, without a doubt, in terms of symmetry, is an unsurpassed embodiment of perfection, an ideal. The ancient Greeks perceived it as the most perfect body, and the circle, naturally, as the most perfect flat figure.

To describe the symmetry of a particular object, it is necessary to indicate all the rotation axes and their order, as well as all planes of symmetry.

Consider, for example, geometric body, composed of two identical regular quadrangular pyramids.

It has one rotary axis of the 4th order (axis AB), four rotary axes of the 2nd order (axes CE,DF, MP, NQ), five planes of symmetry (planesCDEF, AFBD, ACBE, AMBP, ANBQ).

5 . Portable symmetry

Another type of symmetry isportable With symmetry.

Such symmetry is said to occur when, when moving a figure along a straight line to some distance “a” or a distance that is a multiple of this value, it aligns with itself The straight line along which the transfer occurs is called the transfer axis, and the distance “a” is called the elementary transfer, period or symmetry step.

A

A periodically repeating pattern on a long strip is called a border. In practice, borders are found in various forms (wall painting, cast iron, plaster bas-reliefs or ceramics). Borders are used by painters and artists when decorating a room. To make these ornaments, a stencil is made. We move the stencil, turning it over or not, tracing the outline, repeating the pattern, and we get an ornament (visual demonstration).

The border is easy to build using a stencil (the starting element), moving or turning it over and repeating the pattern. The figure shows five types of stencils:A ) asymmetrical;b, c ) having one axis of symmetry: horizontal or vertical;G ) centrally symmetrical;d ) having two axes of symmetry: vertical and horizontal.

To construct borders, the following transformations are used:

A ) parallel transfer;b ) symmetry about the vertical axis;V ) central symmetry;G ) symmetry about the horizontal axis.

You can build sockets in the same way. To do this, the circle is divided inton equal sectors, in one of them a sample pattern is made and then the latter is sequentially repeated in the remaining parts of the circle, rotating the pattern each time by an angle of 360°/n .

A clear example For the application of axial and portable symmetry, the fence shown in the photograph can serve.

Conclusion: Thus, there are different types of symmetry, symmetrical points in each of these types of symmetry are constructed according to certain laws. In life, we encounter one type of symmetry everywhere, and often in the objects that surround us, several types of symmetry can be noted at once. This creates order, beauty and perfection in the world around us.

LITERATURE:

Handbook of Elementary Mathematics. M.Ya. Vygodsky. – Publishing house “Nauka”. – Moscow 1971 – 416 pages.

Modern dictionary foreign words. - M.: Russian language, 1993.

History of mathematics in schoolIX - Xclasses. G.I. Glaser. – Publishing house “Prosveshcheniye”. – Moscow 1983 – 351 pages.

Visual geometry 5th – 6th grades. I.F. Sharygin, L.N. Erganzhieva. – Publishing house “Drofa”, Moscow 2005. – 189 pages

Encyclopedia for children. Biology. S. Ismailova. – Avanta+ Publishing House. – Moscow 1997 – 704 pages.

Urmantsev Yu.A. Symmetry of nature and the nature of symmetry - M.: Mysl arxitekt / arhkomp2. htm, , ru.wikipedia.org/wiki/

Today we will talk about a phenomenon that each of us constantly encounters in life: symmetry. What is symmetry?

We all roughly understand the meaning of this term. The dictionary says: symmetry is proportionality and complete correspondence of the arrangement of parts of something relative to a straight line or point. There are two types of symmetry: axial and radial. Let's look at the axial one first. This is, let’s say, “mirror” symmetry, when one half of an object is completely identical to the second, but repeats it as a reflection. Look at the halves of the sheet. They are mirror symmetrical. The halves of the human body are also symmetrical (front view) - identical arms and legs, identical eyes. But let’s not be mistaken; in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet copy each other far from perfectly, the same applies to human body(take a closer look for yourself); The same is true for other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer only in one position. It’s worth, say, turning a sheet of paper, or raising one hand, and what happens? – you see for yourself.

People achieve true symmetry in the works of their labor (things) - clothes, cars... In nature, it is characteristic of inorganic formations, for example, crystals.

But let's move on to practice. It’s not worth starting with complex objects like people and animals; let’s try, as the first exercise in a new field, to finish drawing the mirror half of the sheet.

Drawing a symmetrical object - lesson 1

We make sure that it turns out as similar as possible. To do this, we will literally build our soul mate. Don’t think that it’s so easy, especially the first time, to draw a mirror-corresponding line with one stroke!

Let's mark several reference points for the future symmetrical line. We proceed like this: with a pencil, without pressing, we draw several perpendiculars to the axis of symmetry - the midrib of the leaf. Four or five is enough for now. And on these perpendiculars we measure to the right the same distance as on the left half to the line of the edge of the leaf. I advise you to use a ruler, don’t rely too much on your eye. As a rule, we tend to reduce the drawing - this has been observed from experience. We do not recommend measuring distances with your fingers: the error is too large.

Let's connect the resulting points with a pencil line:

Now let’s look meticulously to see if the halves are really the same. If everything is correct, we will circle it with a felt-tip pen and clarify our line:

The poplar leaf has been completed, now you can take a swing at the oak leaf.

Let's draw a symmetrical figure - lesson 2

In this case, the difficulty lies in the fact that the veins are marked and they are not perpendicular to the axis of symmetry and not only the dimensions but also the angle of inclination will have to be strictly observed. Well, let’s train our eye:

So a symmetrical oak leaf has been drawn, or rather, we built it according to all the rules:

How to draw a symmetrical object - lesson 3

And let’s consolidate the theme - we’ll finish drawing a symmetrical lilac leaf.

He has too interesting shape- heart-shaped and with ears at the base, you’ll have to puff:

Here's what they drew:

Take a look at the resulting work from a distance and evaluate how accurately we were able to convey the required similarity. Here's a tip: look at your image in the mirror and it will tell you if there are mistakes. Another way: bend the image exactly along the axis (we have already learned how to bend it correctly) and cut out the leaf along the original line. Look at the figure itself and at the cut paper.

I . Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, whenmeasures)

Summary table (all properties, features)

II . Applications of symmetry:

1) in mathematics

2) in chemistry

3) in biology, botany and zoology

4) in art, literature and architecture

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1. Basic concepts of symmetry and its types.

The concept of symmetry r goes back through the entire history of mankind. It is found already at the origins of human knowledge. It arose in connection with the study of a living organism, namely man. And it was used by sculptors back in the 5th century BC. e. The word “symmetry” is Greek and means “proportionality, proportionality, sameness in the arrangement of parts.” It is widely used by all areas of modern science without exception. Many great people have thought about this pattern. For example, L.N. Tolstoy said: “Standing in front of a black board and drawing different figures on it with chalk, I was suddenly struck by the thought: why is symmetry clear to the eye? What is symmetry? This is an innate feeling, I answered myself. What is it based on?” The symmetry is truly pleasing to the eye. Who hasn’t admired the symmetry of nature’s creations: leaves, flowers, birds, animals; or human creations: buildings, technology, everything that surrounds us since childhood, everything that strives for beauty and harmony. Hermann Weyl said: “Symmetry is the idea through which man throughout the ages has tried to comprehend and create order, beauty and perfection.” Hermann Weyl is a German mathematician. His activities span the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what signs the presence or, conversely, absence of symmetry in a given case can be discerned. Thus, a mathematically rigorous concept was formed relatively recently - at the beginning of the twentieth century. It's quite complicated. Let us turn and once again remember the definitions that were given to us in the textbook.

2. Axial symmetry.

2.1 Basic definitions

Definition. Two points A and A 1 are called symmetrical with respect to line a if this line passes through the middle of segment AA 1 and is perpendicular to it. Each point of the line a is considered symmetrical to itself.

Definition. The figure is said to be symmetrical about a straight line A, if for each point of the figure there is a point symmetrical to it relative to the straight line A also belongs to this figure. Straight A called the axis of symmetry of the figure. The figure is also said to have axial symmetry.

2.2 Construction plan

And so, to construct a symmetrical figure relative to a straight line, from each point we draw a perpendicular to this straight line and extend it by the same distance, mark the resulting point. We do this with each point and get symmetrical vertices of a new figure. Then we connect them in series and get a symmetrical figure of this relative axis.

2.3 Examples of figures with axial symmetry.

3. Central symmetry

3.1 Basic definitions

Definition. Two points A and A 1 are called symmetrical with respect to point O if O is the middle of the segment AA 1. Point O is considered symmetrical to itself.

Definition. A figure is said to be symmetrical with respect to point O if, for each point of the figure, a point symmetrical with respect to point O also belongs to this figure.

3.2 Construction plan

Construction of a triangle symmetrical to the given one relative to the center O.

To construct a point symmetrical to a point A relative to the point ABOUT, it is enough to draw a straight line OA(Fig. 46 ) and on the other side of the point ABOUT set aside a segment equal to the segment OA. In other words , points A and ; In and ; C and symmetrical about some point O. In Fig. 46 a triangle is constructed that is symmetrical to a triangle ABC relative to the point ABOUT. These triangles are equal.

Construction of symmetrical points relative to the center.

In the figure, points M and M 1, N and N 1 are symmetrical relative to point O, but points P and Q are not symmetrical relative to this point.

In general, figures that are symmetrical about a certain point are equal .

3.3 Examples

Let us give examples of figures that have central symmetry. The simplest figures with central symmetry are the circle and parallelogram.

Point O is called the center of symmetry of the figure. In such cases, the figure has central symmetry. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals.

A straight line also has central symmetry, but unlike a circle and a parallelogram, which have only one center of symmetry (point O in the figure), a straight line has an infinite number of them - any point on the straight line is its center of symmetry.

The pictures show an angle symmetrical relative to the vertex, a segment symmetrical to another segment relative to the center A and a quadrilateral symmetrical about its vertex M.

An example of a figure that does not have a center of symmetry is a triangle.

4. Lesson summary

Let us summarize the knowledge gained. Today in class we learned about two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.

Summary table

	Axial symmetry	Central symmetry
Peculiarity	All points of the figure must be symmetrical relative to some straight line.	All points of the figure must be symmetrical relative to the point chosen as the center of symmetry.
Properties	1. Symmetrical points lie on perpendiculars to a line. 3. Straight lines turn into straight lines, angles into equal angles. 4. The sizes and shapes of the figures are preserved.	1. Symmetrical points lie on a line passing through the center and this point figures. 2. The distance from a point to a straight line is equal to the distance from a straight line to a symmetrical point. 3. The sizes and shapes of the figures are preserved.

II. Application of symmetry

Mathematics	In algebra lessons we studied the graphs of the functions y=x and y=x The pictures show various pictures depicted using the branches of parabolas. (a) Octahedron, (b) rhombic dodecahedron, (c) hexagonal octahedron.
Russian language	The printed letters of the Russian alphabet also have different types of symmetries. There are “symmetrical” words in the Russian language - palindromes, which can be read equally in both directions.	A D L M P T F W– vertical axis V E Z K S E Y - horizontal axis F N O X- both vertical and horizontal B G I Y R U C CH SCHY- no axis Radar hut Alla Anna
Literature	Sentences can also be palindromic. Bryusov wrote a poem “The Voice of the Moon”, in which each line is a palindrome. Look at the quadruples of A.S. Pushkin “ Bronze Horseman" If we draw a line after the second line we can notice elements of axial symmetry	And the rose fell on Azor's paw. I come with the sword of the judge. (Derzhavin) "Search for a taxi" "Argentina beckons the Negro" “The Argentinean appreciates the black man,” “Lesha found a bug on the shelf.” The Neva is dressed in granite; Bridges hung over the waters; Dark green gardens Islands covered it...

Biology

The human body is built on the principle of bilateral symmetry. Most of us view the brain as a single structure; in reality, it is divided into two halves. These two parts - two hemispheres - fit tightly to each other. In full accordance with the general symmetry of the human body, each hemisphere is an almost exact mirror image of the other Control of the basic movements of the human body and its sensory functions is evenly distributed between the two hemispheres of the brain. The left hemisphere controls the right side of the brain, and the right hemisphere controls the left side.

Botany

A flower is considered symmetrical when each perianth consists of an equal number of parts. Flowers having paired parts are considered flowers with double symmetry, etc. Triple symmetry is common in monocotyledons, and quintuple symmetry in dicotyledons. Characteristic feature The structure of plants and their development is helicity. Pay attention to the shoots of leaf arrangement - this is also peculiar appearance spiral - helical. Even Goethe, who was not only a great poet, but also a natural scientist, considered helicity one of characteristic features of all organisms, a manifestation of the innermost essence of life. The tendrils of plants twist in a spiral, the growth of tissues in tree trunks occurs in a spiral, the seeds in a sunflower are arranged in a spiral, and spiral movements are observed during the growth of roots and shoots.

A characteristic feature of the structure of plants and their development is spirality. Look at the pine cone. The scales on its surface are arranged strictly regularly - along two spirals that intersect approximately at a right angle. The number of such spirals is pine cones equals 8 and 13 or 13 and 21.

Zoology

Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides of the dividing line. With radial or radial symmetry, the body has the shape of a short or long cylinder or vessel with a central axis, from which parts of the body extend radially. These are coelenterates, echinoderms, starfish. With bilateral symmetry, there are three axes of symmetry, but only one pair of symmetrical sides. Because the other two sides - abdominal and dorsal - are not similar to each other. This type of symmetry is characteristic of most animals, including insects, fish, amphibians, reptiles, birds, and mammals.

Axial symmetry

	Various types symmetry of physical phenomena: symmetry of electric and magnetic fields (Fig. 1) The distribution is symmetrical in mutually perpendicular planes electromagnetic waves(Fig. 2)	Fig.1 Fig.2
Art	Mirror symmetry can often be observed in works of art. Mirror" symmetry is widely found in works of art of primitive civilizations and in ancient paintings. Medieval religious paintings are also characterized by this type of symmetry. One of Raphael’s best early works, “The Betrothal of Mary,” was created in 1504. Under a sunny blue sky lies a valley topped by a white stone temple. In the foreground is the betrothal ceremony. The High Priest brings Mary and Joseph's hands together. Behind Mary is a group of girls, behind Joseph is a group of young men. Both parts of the symmetrical composition are held together by the counter-movement of the characters. For modern tastes, the composition of such a painting is boring, since the symmetry is too obvious.

Chemistry	A water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the world of living nature. It is a double-chain high-molecular polymer, the monomer of which is nucleotides. DNA molecules have a double helix structure built on the principle of complementarity.
Architeculture	Man has long used symmetry in architecture. The ancient architects made especially brilliant use of symmetry in architectural structures. Moreover, the ancient Greek architects were convinced that in their works they were guided by the laws that govern nature. By choosing symmetrical forms, the artist thereby expressed his understanding of natural harmony as stability and balance. The city of Oslo, the capital of Norway, has an expressive ensemble of nature and art. This is Frogner - a park - a complex of garden and park sculpture, which was created over the course of 40 years.	Pashkov House Louvre (Paris)

© Sukhacheva Elena Vladimirovna, 2008-2009.

In this lesson we will look at another characteristic of some figures - axial and central symmetry. We encounter axial symmetry every day when we look in the mirror. Central symmetry is very common in living nature. At the same time, figures that have symmetry have a number of properties. In addition, we subsequently learn that axial and central symmetries are types of movements with the help of which a whole class of problems is solved.

This lesson dedicated to axial and central symmetry.

Definition

The two points are called symmetrical relatively straight if:

In Fig. 1 shows examples of points symmetrical with respect to a straight line and , and .

Rice. 1

Let us also note the fact that any point on a line is symmetrical to itself relative to this line.

Figures can also be symmetrical relative to a straight line.

Let us formulate a strict definition.

Definition

The figure is called symmetrical relative to straight, if for each point of the figure the point symmetrical to it relative to this straight line also belongs to the figure. In this case the line is called axis of symmetry. The figure has axial symmetry.

Let's look at a few examples of figures that have axial symmetry and their axes of symmetry.

Example 1

The angle has axial symmetry. The axis of symmetry of the angle is the bisector. Indeed: let’s lower a perpendicular to the bisector from any point of the angle and extend it until it intersects with the other side of the angle (see Fig. 2).

Rice. 2

(since - the common side, (property of a bisector), and triangles are right-angled). Means, . Therefore, the points are symmetrical with respect to the bisector of the angle.

It follows from this that an isosceles triangle also has axial symmetry with respect to the bisector (altitude, median) drawn to the base.

Example 2

An equilateral triangle has three axes of symmetry (bisectors/medians/altitudes of each of the three angles (see Fig. 3).

Rice. 3

Example 3

A rectangle has two axes of symmetry, each of which passes through the midpoints of its two opposite sides (see Fig. 4).

Rice. 4

Example 4

A rhombus also has two axes of symmetry: straight lines, which contain its diagonals (see Fig. 5).

Rice. 5

Example 5

A square, which is both a rhombus and a rectangle, has 4 axes of symmetry (see Fig. 6).

Rice. 6

Example 6

For a circle, the axis of symmetry is any straight line passing through its center (that is, containing the diameter of the circle). Therefore, a circle has infinitely many axes of symmetry (see Fig. 7).

Rice. 7

Let us now consider the concept central symmetry.

Definition

The points are called symmetrical relative to the point if: - the middle of the segment.

Let's look at a few examples: in Fig. 8 shows the points and , as well as and , which are symmetrical with respect to the point , and the points and are not symmetrical with respect to this point.

Rice. 8

Some figures are symmetrical about a certain point. Let us formulate a strict definition.

Definition

The figure is called symmetrical about the point, if for any point of the figure the point symmetrical to it also belongs to this figure. The point is called center of symmetry, and the figure has central symmetry.

Let's look at examples of figures with central symmetry.

Example 7

For a circle, the center of symmetry is the center of the circle (this is easy to prove by recalling the properties of the diameter and radius of a circle) (see Fig. 9).

Rice. 9

Example 8

For a parallelogram, the center of symmetry is the point of intersection of the diagonals (see Fig. 10).

Rice. 10

Let's solve several problems on axial and central symmetry.

Task 1.

How many axes of symmetry does the segment have?

A segment has two axes of symmetry. The first of them is a line containing a segment (since any point on a line is symmetrical to itself relative to this line). The second is the perpendicular bisector to the segment, that is, a straight line perpendicular to the segment and passing through its middle.

Answer: 2 axes of symmetry.

Task 2.

How many axes of symmetry does a straight line have?

A straight line has infinitely many axes of symmetry. One of them is the line itself (since any point on the line is symmetrical to itself relative to this line). And also the axes of symmetry are any lines perpendicular to a given line.

Answer: there are infinitely many axes of symmetry.

Task 3.

How many axes of symmetry does the beam have?

The ray has one axis of symmetry, which coincides with the line containing the ray (since any point on the line is symmetrical to itself relative to this line).

Answer: one axis of symmetry.

Task 4.

Prove that the lines containing the diagonals of a rhombus are its axes of symmetry.

Proof:

Consider a rhombus. Let us prove, for example, that the straight line is its axis of symmetry. It is obvious that the points are symmetrical to themselves, since they lie on this line. In addition, the points and are symmetrical with respect to this line, since . Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the rhombus (see Fig. 11).

Rice. 11

Draw a perpendicular to the line through the point and extend it until it intersects with . Consider triangles and . These triangles are right-angled (by construction), in addition, they have: - a common leg, and (since the diagonals of a rhombus are its bisectors). So these triangles are equal: . This means that all their corresponding elements are equal, therefore: . From the equality of these segments it follows that the points and are symmetrical with respect to the straight line. This means that it is the axis of symmetry of the rhombus. This fact can be proven similarly for the second diagonal.

Proven.

Task 5.

Prove that the point of intersection of the diagonals of a parallelogram is its center of symmetry.

Proof:

Consider a parallelogram. Let us prove that the point is its center of symmetry. It is obvious that the points and , and are pairwise symmetrical with respect to the point , since the diagonals of a parallelogram are divided in half by the point of intersection. Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the parallelogram (see Fig. 12).

Goals:

• educational:
  • give an idea of ​​symmetry;
  • introduce the main types of symmetry on the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand your understanding of famous figures by introducing properties associated with symmetry;
  • show the possibilities of using symmetry in solving various problems;
  • consolidate acquired knowledge;
• general education:
  • teach yourself how to prepare yourself for work;
  • teach how to control yourself and your desk neighbor;
  • teach to evaluate yourself and your desk neighbor;
• developing:
  • intensify independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • develop a “shoulder sense” in students;
  • cultivate communication skills;
  • instill a culture of communication.

PROGRESS OF THE LESSON

In front of each person are scissors and a sheet of paper.

Task 1(3 min).

- Let's take a sheet of paper, fold it into pieces and cut out some figure. Now let's unfold the sheet and look at the fold line.

Question: What function does this line serve?

Suggested answer: This line divides the figure in half.

Question: How are all the points of the figure located on the two resulting halves?

Suggested answer: All points of the halves are on equal distance from the fold line and at the same level.

– This means that the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is an axis of symmetry.

Task 2 (2 min).

– Cut out a snowflake, find the axis of symmetry, characterize it.

Task 3 (5 min).

– Draw a circle in your notebook.

Question: Determine how the axis of symmetry goes?

Suggested answer: Differently.

Question: So how many axes of symmetry does a circle have?

Suggested answer: Many.

– That’s right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)

Question: What other figures have more than one axis of symmetry?

Suggested answer: Square, rectangle, isosceles and equilateral triangles.

– Consider three-dimensional figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry do the square, rectangle, equilateral triangle and the proposed three-dimensional figures have?

I distribute halves of plasticine figures to students.

Task 4 (3 min).

– Using the information received, complete the missing part of the figure.

Note: the figure can be both planar and three-dimensional. It is important that students determine how the axis of symmetry runs and complete the missing element. The correctness of the work is determined by the neighbor at the desk and evaluates how correctly the work was done.

A line (closed, open, with self-intersection, without self-intersection) is laid out from a lace of the same color on the desktop.

Task 5 (group work 5 min).

– Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.

The correctness of the work performed is determined by the students themselves.

Elements of drawings are presented to students

Task 6 (2 min).

– Find the symmetrical parts of these drawings.

To consolidate the material covered, I suggest the following tasks, scheduled for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What type of triangles are these?

2. Draw several isosceles triangles in your notebook with a common base of 6 cm.

3. Draw a segment AB. Construct a line segment AB perpendicular and passing through its midpoint. Mark points C and D on it so that the quadrilateral ACBD is symmetrical with respect to the straight line AB.

– Our initial ideas about form date back to the very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new Stone Age, in the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
– Where does symmetry occur in nature?

Suggested answer: wings of butterflies, beetles, tree leaves...

– Symmetry can also be observed in architecture. When constructing buildings, builders strictly adhere to symmetry.

That's why the buildings turn out so beautiful. Also an example of symmetry is humans and animals.

Homework:

1. Come up with your own ornament, draw it on an A4 sheet (you can draw it in the form of a carpet).
2. Draw butterflies, mark where elements of symmetry are present.