How to construct a figure symmetrical to a given one with respect to. Central and axial symmetry

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

Approximately we all understand the meaning of this term. The dictionary says: symmetry is the proportionality and full correspondence of the arrangement of parts of something relative to a line or point. There are two types of symmetry: axial and radial. Let's look at the axis first. This is, let's say, "mirror" symmetry, when one half of the 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 (full face) are also symmetrical - the same arms and legs, the same eyes. But let's not be mistaken, in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet do not copy each other perfectly, the same applies to human body(take a look yourself); the same is true of other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer in only one position. It is necessary, say, to turn the sheet, or raise one hand, and what? - see for yourself.

People achieve true symmetry in the products 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 to finish the mirror half of the sheet as the first exercise in a new field.

Draw a symmetrical object - lesson 1

Let's try to make it as similar as possible. To do this, we will literally build our soul mate. Do not think that it is 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 act like this: we draw with a pencil without pressure several perpendiculars to the axis of symmetry - the middle vein of the sheet. Four or five is enough. 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 the ruler, do not really rely on the eye. As a rule, we tend to reduce the drawing - it has been noticed in experience. We do not recommend measuring distances with your fingers: the error is too large.

Connect the resulting points with a pencil line:

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

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

Let's draw a symmetrical figure - lesson 2

In this case, the difficulty lies in the fact that the veins are indicated 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 exactly observed. Well, let's train the eye:

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

How to draw a symmetrical object - lesson 3

And we will fix the topic - we will finish drawing a symmetrical leaf of lilac.

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

Here is what they drew:

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

Goals:

• educational:
  • give an idea of ​​\u200b\u200bsymmetry;
  • introduce the main types of symmetry in the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand ideas about famous figures by introducing them to the properties associated with symmetry;
  • show the possibilities of using symmetry in solving various problems;
  • consolidate the acquired knowledge;
• general education:
  • learn to set yourself up for work;
  • teach to control oneself and a neighbor on the desk;
  • to teach how to evaluate yourself and a neighbor on your desk;
• developing:
  • activate independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • educate students "a sense of shoulder";
  • cultivate communication;
  • inculcate a culture of communication.

DURING THE CLASSES

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

Exercise 1(3 min).

- Take a sheet of paper, fold it in half and cut out some figure. Now unfold the sheet and look at the fold line.

Question: What is the function of this line?

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 at an equal distance from the fold line and at the same level.

- So, 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 the axis of symmetry.

Task 2 (2 minutes).

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

Task 3 (5 minutes).

- Draw a circle in your notebook.

Question: Determine how the axis of symmetry passes?

Suggested answer: Differently.

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

Suggested answer: Lot.

- That's right, the circle has many axes of symmetry. The same wonderful figure is the 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 a square, rectangle, equilateral triangle and the proposed three-dimensional figures have?

I distribute the halves of plasticine figures to the students.

Task 4 (3 min).

- Using the information received, finish the missing part of the figure.

Note: the figurine can be both flat and three-dimensional. It is important that students determine how the axis of symmetry goes and fill in the missing element. The correctness of the execution is determined by the neighbor on the desk, evaluates how well the work has been done.

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

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.

The students are presented with elements of drawings

Task 6 (2 minutes).

Find the symmetrical parts of these drawings.

To consolidate the material covered, I propose the following tasks, provided for 15 minutes:

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

2. Draw in a notebook several isosceles triangles with a common base equal to 6 cm.

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

- Our initial ideas about the form belong to a 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 that differed little from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and in the late Paleolithic era, they decorated their existence by creating works of art, figurines and drawings, which reveal a wonderful sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity enters a new stone Age, in the Neolithic.
Neolithic man had a keen sense of geometric form. The firing and coloring of clay vessels, the manufacture of 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 is symmetry found in nature?

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

“Symmetry can also be seen in architecture. When constructing buildings, builders clearly adhere to symmetry.

That's why the buildings are so beautiful. Also an example of symmetry is a person, animals.

Homework:

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

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Lessons at school are a significant part of the life of schoolchildren, requiring elementary comfort, favorable communication. The effectiveness of the educational process depends not only on the abilities of diligence and diligence of students, the presence of purposeful motivation of the teacher, but also on the form of conducting lessons.

Usage information technologies allows you to save time when explaining new material, to present the material in a visual, understandable form, to influence different systems of perception of students, thereby ensuring better assimilation of the material.

Much attention is paid to the application of acquired knowledge in mathematics in everyday life. Acquaintance with beauty in life and art not only educates the mind and feeling of the child, but also contributes to the development of imagination and fantasy. I believe that a lesson with elements creative activity helps to activate the mental activity of schoolchildren and therefore takes place at a high emotional level, which allows us to consider a large number of theoretical issues and tasks, to involve all students in the class. In order to increase the activity of students throughout the lesson, alternation of activities is used.

At the end of the lesson, students do verification work in the form of a test, they conduct a self-examination, evaluating their work according to specified criteria. The most active group of students was offered additional material on topics studied.

Reflection at the end of the lesson helps to determine the level of assimilation of the material and set goals for further work.

Homework consists of two parts, which allows not only to continue to consolidate the acquired knowledge, but to develop the creative abilities of children.

In my opinion, such lessons enable the teacher to create, search, work for high results, form students' universal learning activities– thus preparing them for continuing education and for life in an ever-changing environment.

Lesson Objectives:

• familiarity with the concept of axial symmetry;
• the formation of skills to build symmetrical figures with respect to a straight line and to identify axial symmetry as a property of some geometric shapes;
• revealing the links between mathematics and wildlife, art, technology, architecture;
• development of skills to apply knowledge of theory in practice, development of skills of self-control and mutual control, self-assessment and introspection learning activities;
• development of attention, observation, thinking, interest in the subject, mathematical speech, the desire for creativity;
• formation of aesthetic perception of the surrounding world, education of independence.
• preparing students for the study of geometry, deepening existing knowledge;

Lesson type: lesson of "discovery" of new knowledge.

Equipment: computer, pin or compass, projector, cards, paper geometric shapes.

DURING THE CLASSES

1. Organizing moment

(Slide 1) It is easy to find examples of beauty, but how difficult it is to explain why they are beautiful. (Plato)

– Today in the lesson we will try to understand some of the features of creating beauty!!!

2. Update

- Look at the maple leaf, snowflake, butterfly. (Slide 2) What unites them, what do they have in common? That they are symmetrical.
Remind me, please, what the word "symmetry" means.
- "Symmetry" in Greek means "proportionality, proportionality, the sameness in the arrangement of parts." If you put a mirror along the straight line drawn in each figure, then the half of the figure reflected on the mirror will complement it to the whole. Therefore, such symmetry is called mirror (axial).

(The teacher shows the experience on a Christmas tree cut out of colored paper)

The straight line along which the mirror is placed is called axis of symmetry. If you bend the sheet along this straight line, then these figures fully match, and we can see only one figure. What do you think is the topic of today's lesson? (Axial symmetry)

(Slides 3-4)

- Guys, today we will learn how to build symmetrical figures with respect to a straight line, and you will also learn where axial symmetry is applied.
How do you get symmetrical shapes?
- First, consider the easiest way to get symmetrical shapes.
Each of you has a piece of white paper on your desk. Take a piece of paper and fold it in half. Now on the same side build a triangle(Row 1 is acute-angled, row 2 is rectangular, row 3 is obtuse).
Further pierce tops of this figure so that both halves are pierced. Now expand the sheet and connect the resulting dots-holes along the ruler. Thus, we have built figures that are symmetrical to the data with respect to a straight line (inflection line). Check it out. To do this, fold the sheet along the fold line and look through it at the light.
– What do you see? (The figures matched.)
– This is the easiest way to build symmetrical figures.
- But is it always in practice, in this way, that we can build symmetrical figures?
– And what have we done in order to build symmetrical triangles?
- Fold the paper in half.
- I.e. draw an axis of symmetry. Farther.
- Pierced the vertices of the triangle.
- I.e. built the points by which our triangle is bounded.
- And this means that before constructing a figure symmetrical to a given one, we must learn to build first of all what? (A point symmetrical to a given one.)
- How can this be done, let's figure it out.

3. Let's do it now practical work:

- Mark a point Ah. From a point BUT lower the perpendicular JSC directly a. Now from the point O set aside a perpendicular OA1= AO. Two dots BUT and A1 called symmetrical with respect to the line a. This line is called the axis of symmetry.

(The teacher builds on the board, students in notebooks).

What two points are said to be symmetrical about a line?
- And how to construct a figure symmetrical about some line?
- Let's try to build a triangle symmetrical about a straight line.

(The teacher calls the willing student to the board, the rest work in notebooks).

After the work done, the students make a conclusion together with the teacher.

Conclusion: To construct a geometric figure symmetrical to a given one with respect to some straight line, one must build points, symmetrical to significant points ( peaks) of this figure with respect to this line and then connect these points with lines.

- Guys, symmetrical can be not only 2 figures, in some figures You can also draw an axis of symmetry. Such figures are said to have axial symmetry. Name the figures that have axial symmetry.

(The teacher names and shows geometric shapes cut out of colored paper)

How many axes of symmetry do you think isosceles triangle, rectangle, square? (Rectangle has 2 axes of symmetry. Square has 4 axes of symmetry) – And at the circle? (A circle has infinitely many axes of symmetry).

(Slides 7-11)

- Name the figures that do not have an axis of symmetry. (Parallelogram, scalene triangle, irregular polygon).

– The principles of symmetry play important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music. Almost all vehicles, household items (furniture, dishes), some musical instruments are symmetrical.
– Give examples of objects having axial symmetry.

– Nature laws, controlling the inexhaustible in its diversity picture of the phenomenon, in turn, also obey the principles of symmetry. Careful observation shows that the basis of the beauty of many forms created by nature is symmetry.

(Slides 12-15)

Symmetry is often found in man-made objects.
Symmetry is found already at the origins of human development. From time immemorial, man has used symmetry in architecture. Ancient temples, towers of medieval castles, modern buildings it gives harmony, completeness.

(Slides 18-19)

Impressive results are given by symmetry in the fine arts. (Slides 20-21)
Renaissance artists often used the language of symmetry in the construction of their compositions. This followed from their logic of understanding the picture as an image of an ideal world order, where reasonable organization and balance reign, which a person can cognize and comprehend.
In amazing painting "Betrothal of the Virgin Mary" great Raphael reproduced such an image of the world that exists according to the laws of harmony and strict logic. The principle of symmetry used creates the impression of peace and solemnity and at the same time a certain detachment from the viewer. The entrance to the graceful rotunda and the ring put on by Joseph on Mary's hand coincide with the central axis of symmetry of the painting.
In work Leonardo "The Last Supper" strict constructions of the interior perspective prevail. The compositional development here is based on the mirror repetition of the right and left parts. Of course, most often in the visual arts we say about incomplete symmetry.
in the picture "Three heroes" by Russian artist V. Vasnetsov the characters themselves are full of pent-up power. Because of these small deviations from strict symmetry, there is a feeling of inner freedom of the characters, their readiness for movement.
The letters of the Russian language can also be considered from the point of view of symmetry. (Slides 22-23)
The whole alphabet is divided into 4 groups, what do you think, by what criteria did I do this?
The letters A, M, T, W, P have a vertical axis of symmetry, B, Z, K, C, E, B, E - horizontal. And the letters Zh, N, O, F, X have two axes of symmetry.
Symmetry can also be seen in the words: Cossack, hut. There are whole phrases with this property (if you do not take into account the spaces between words): “Look for a taxi”, “Argentina beckons a black man”, “Argentine appreciates a black man”. Such words are called palindromes . Many poets were fond of them.
Consider examples of words that have a horizontal axis of symmetry:
SNOW, CALL, HORSE, NOSE
Words that have a vertical axis of symmetry:

X	T
O	O
L	P
O	O
D	T

Some composers, including the great Bach, wrote musical palindromes.

(Slide 24) Those who are lucky enough to have a symmetrical face have probably already noticed that they are popular with the opposite sex. It may also indicate their good health. The point is that the face perfect proportions is a sign that the body of its owner is well prepared to fight infections. The common cold, asthma, and flu are highly likely to recede in front of people whose left side is exactly like the right.

Physical education minute(Slide 25)

One - rise, stretch,
Two - bend, unbend.
Three - in the hands of three claps,
Tori nods his head.
Four - arms wider,
Five - wave your hands,
Six - sit down at the desk again.

(Slide 26-27)

A test is carried out followed by a self-test.

- Let's not forget about the gymnastics of the mind. Our examples today are also symmetrical. Who has already completed the task, you can count these symmetrical examples orally. (Slide 30)

Option 1 Option 2

1) B 2) D 3) B 4) A 5) C 1) C 2) B 3) B 4) D 5) D

Evaluation of the work performed according to the relevant criteria:

"5" - 5 tasks;
"4" - 4 tasks;
"3" - 3 tasks;
"2" - less than three tasks.

- Try to answer the question which figure is superfluous and why? (Slide 31)

(Figure No. 3, because it does not have an axis of symmetry)

- Well done!

5. The result of the lesson. Reflection

- Our lesson is coming to an end, but the acquaintance with symmetry continues. Throughout the lesson, we performed a variety of tasks.
What concept did you meet today?
What are the goals for the lesson? Have we achieved our goals? Who worked the best? Who excelled in class? What task did you find the most difficult? What theoretical material helped to cope with the task?
What task did you find most interesting? What new things did you “discover” during the lesson? What do you think each of you should work on?

Guys, thank you for your work! Without each other's help and support, we would not have been able to reach our goal. I am very pleased with your work in class. Do you think that we did not spend these moments together in vain? Share your impressions about our lesson.

(Slides 32-33)

7. Conclusion

Truly symmetrical objects surround us literally from all sides, we are dealing with symmetry wherever there is any order. Symmetry resists chaos, disorder. It turns out that symmetry is balance, orderliness, beauty, perfection.
The whole world can be considered as a manifestation of the unity of symmetry and asymmetry. Symmetry is manifold, ubiquitous. She creates beauty and harmony.
And to the question: “Is there a future without symmetry?” we can answer with the words of the classic of modern natural science, the thinker Vladimir Ivanovich Vernadsky “The principle of symmetry covers more and more new areas ...”

I . Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, withmeasures)

Summary table (all properties, features)

II . Symmetry Applications:

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 n R runs throughout the 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 as early as the 5th century BC. e. The word "symmetry" is Greek, it means "proportionality, proportionality, the sameness in the arrangement of parts." It is widely used by all areas of modern science without exception. Many great people 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 really pleasing to the eye. Who has not admired the symmetry of nature's creations: leaves, flowers, birds, animals; or human creations: buildings, technology, - all that surrounds us from childhood, that strives for beauty and harmony. Hermann Weyl said: "Symmetry is the idea through which man has tried for centuries to comprehend and create order, beauty and perfection." Hermann Weyl is a German mathematician. Its activity falls on the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what signs to see the presence or, conversely, the absence of symmetry in a particular case. Thus, a mathematically rigorous representation was formed relatively recently - at the beginning of the 20th century. It is quite complex. We will turn and once again recall the definitions that are 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 the line a if this line passes through the midpoint of the 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 with respect to a straight line. a, if for each point of the figure the point symmetrical to it with respect 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 build 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, we get the symmetrical vertices of the 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 the point O if O is the midpoint of the segment AA 1. Point O is considered symmetrical to itself.

Definition. A figure is called symmetric with respect to the point O if for each point of the figure the point symmetric to it with respect to the point O also belongs to this figure.

3.2 Construction plan

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

To construct a point symmetrical to a point BUT relative to the point O, it suffices to draw a straight line OA(Fig. 46 ) and on the other side of the point O set aside a segment equal to a segment OA. In other words , points A and ; In and ; C and are symmetrical with respect to some point O. In fig. 46 built a triangle symmetrical to a triangle ABC relative to the point O. These triangles are equal.

Construction of symmetrical points about the center.

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

In general, figures that are symmetrical about some point are equal to .

3.3 Examples

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

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

The figures show an angle symmetrical about the vertex, a segment symmetrical to another segment about the center BUT 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. Summary of the lesson

Let's summarize the knowledge gained. Today in the lesson we got acquainted with 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 with respect to some straight line.	All points of the figure must be symmetrical about the point chosen as the center of symmetry.
Properties	1. Symmetric points lie on perpendiculars to the line. 3. Straight lines turn into straight lines, angles into equal angles. 4. The sizes and shapes of the figures are saved.	1. Symmetrical points lie on a straight line passing through the center and the given point of the figure. 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 saved.

II. Application of symmetry

© Sukhacheva Elena Vladimirovna, 2008-2009

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Lesson type: combined.

Lesson Objectives:

• Consider axial, central and mirror symmetries as properties of some geometric shapes.
• Learn to build symmetrical points and recognize shapes that have axial symmetry and central symmetry.
• Improve problem solving skills.

Lesson objectives:

• Formation of spatial representations of students.
• Developing the ability to observe and reason; development of interest in the subject through the use of information technology.
• Raising a person who knows how to appreciate the beautiful.

Lesson equipment:

• Use of information technologies (presentation).
• Drawings.
• Homework cards.

During the classes

I. Organizational moment.

Inform the topic of the lesson, formulate the objectives of the lesson.

II. Introduction.

What is symmetry?

The outstanding mathematician Hermann Weyl highly appreciated the role of symmetry in modern science: "Symmetry, no matter how broadly or narrowly we understand this word, is an idea with which a person tried to explain and create order, beauty and perfection."

We live in a very beautiful and harmonious world. We are surrounded by objects that please the eye. For example, a butterfly, a maple leaf, a snowflake. Look how beautiful they are. Did you pay attention to them? Today we will touch this beautiful mathematical phenomenon - symmetry. Let's get acquainted with the concept of axial, central and mirror symmetries. We will learn to build and define figures that are symmetrical about the axis, center and plane.

The word "symmetry" in Greek sounds like "harmony", meaning beauty, proportionality, proportionality, uniformity in the arrangement of parts. Since ancient times, man has used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, modern buildings.

In the most general view"symmetry" in mathematics is understood as such a transformation of space (plane), in which each point M goes to another point M "relative to some plane (or line) a, when the segment MM" is perpendicular to the plane (or line) a and is divided by it in half . The plane (straight line) a is called the plane (or axis) of symmetry. The fundamental concepts of symmetry include the plane of symmetry, the axis of symmetry, the center of symmetry. The plane of symmetry P is a plane that divides the figure into two mirror equal parts, located relative to each other in the same way as an object and its mirror reflection.

III. Main part. Symmetry types.

Central symmetry

Symmetry about a point or central symmetry is such a property geometric figure when any point located on one side of the center of symmetry corresponds to another point located on the other side of the center. In this case, the points are on a straight line segment passing through the center, dividing the segment in half.

Practical task.

1. Given points BUT, AT and M M relative to the middle of the segment AB.
2. Which of the following letters have a center of symmetry: A, O, M, X, K?
3. Do they have a center of symmetry: a) a segment; b) beam; c) a pair of intersecting lines; d) square?

Axial symmetry

Symmetry with respect to a straight line (or axial symmetry) is such a property of a geometric figure, when any point located on one side of the straight line will always correspond to a point located on the other side of the straight line, and the segments connecting these points will be perpendicular to the axis of symmetry and divide it in half.

Practical task.

1. Given two points BUT and AT, symmetric with respect to some straight line, and a point M. Construct a point symmetrical to a point M about the same line.
2. Which of the following letters have an axis of symmetry: A, B, D, E, O?
3. How many axes of symmetry does: a) a segment; b) straight line; c) beam?
4. How many axes of symmetry does the drawing have? (see fig. 1)

Mirror symmetry

points BUT and AT are called symmetric with respect to the plane α (plane of symmetry) if the plane α passes through the midpoint of the segment AB and perpendicular to this segment. Each point of the plane α is considered symmetrical to itself.

Practical task.

1. Find the coordinates of the points into which the points A (0; 1; 2), B (3; -1; 4), C (1; 0; -2) pass with: a) central symmetry about the origin; b) axial symmetry about the coordinate axes; c) mirror symmetry with respect to coordinate planes.
2. Does the right glove go into the right or left glove with mirror symmetry? axial symmetry? central symmetry?
3. The figure shows how the number 4 is reflected in two mirrors. What will be seen in place of the question mark if the same is done with the number 5? (see fig. 2)
4. The figure shows how the word KANGAROO is reflected in two mirrors. What happens if you do the same with the number 2011? (see fig. 3)

Rice. 2

It is interesting.

Symmetry in nature.

Almost all living beings are built according to the laws of symmetry, not without reason, translated from Greek word"symmetry" means "proportionality".

Among colors, for example, rotational symmetry is observed. Many flowers can be rotated so that each petal takes the position of its neighbor, the flower is aligned with itself. The minimum angle of such a rotation for different colors is not the same. For iris, it is 120°, for bluebell - 72°, for narcissus - 60°.

In the arrangement of leaves on the stems of plants, helical symmetry is observed. Being located as a screw along the stem, the leaves, as it were, spread out in different directions and do not block each other from the light, although the leaves themselves also have an axis of symmetry. Considering the general plan of the structure of any animal, we usually notice a well-known regularity in the arrangement of parts of the body or organs that repeat around a certain axis or occupy the same position in relation to a certain plane. This correctness is called the symmetry of the body. The phenomena of symmetry are so widespread in the animal world that it is very difficult to point out a group in which no symmetry of the body can be noticed. Both small insects and large animals have symmetry.

Symmetry in inanimate nature.

Among endless variety forms inanimate nature such perfect images abound in abundance, whose appearance invariably attracts our attention. Observing the beauty of nature, one can notice that when objects are reflected in puddles, lakes, mirror symmetry appears (see Fig. 4).

Crystals bring the charm of symmetry to the world of inanimate nature. Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have rotational symmetry and, in addition, mirror symmetry.

It is impossible not to see the symmetry in faceted gemstones. Many cutters try to shape their diamonds into a tetrahedron, cube, octahedron, or icosahedron. Since the pomegranate has the same elements as the cube, it is highly valued by connoisseurs. precious stones. Artwork made from pomegranates was found in graves ancient egypt dating back to the pre-dynastic period (over two millennia BC) (see Fig. 5).

In the collections of the Hermitage special attention used gold jewelry of the ancient Scythians. Unusually fine art work of gold wreaths, diadems, wood and decorated with precious red-violet garnets.

One of the most obvious uses of the laws of symmetry in life are the structures of architecture. This is what we see most often. In architecture, symmetry axes are used as a means of expressing architectural intent (see Figure 6). In most cases, patterns on carpets, fabrics, and room wallpapers are symmetrical about the axis or center.

Another example of a person using symmetry in his practice is technique. In engineering, axes of symmetry are most clearly indicated where deviation from zero is required, such as on the steering wheel of a truck or on the steering wheel of a ship. Or one of the most important inventions of mankind, having a center of symmetry, is a wheel, also a propeller and other technical means have a center of symmetry.

"Look in the mirror!"

Should we think that we see ourselves only in a "mirror image"? Or, at best, can we find out how we “really” look only on photos and film? Of course not: it is enough to reflect the mirror image a second time in the mirror in order to see your true face. Trills come to the rescue. They have one large main mirror in the center and two smaller mirrors on the sides. If such a side mirror is placed at a right angle to the average, then you can see yourself exactly in the form in which others see you. Close your left eye, and your reflection in the second mirror will repeat your movement with your left eye. Before the trellis, you can choose whether you want to see yourself in a mirror image or in a direct image.

It is easy to imagine what confusion would reign on Earth if the symmetry in nature were broken!

Rice. 4	Rice. 5	Rice. 6

IV. Fizkultminutka.

• « lazy eights» – activate the structures that provide memorization, increase the stability of attention.
  Draw the number eight in the air in a horizontal plane three times, first with one hand, then immediately with both hands.
• « Symmetrical drawings » - improve hand-eye coordination, facilitate the process of writing.
  Draw symmetrical patterns in the air with both hands.

V. Independent work of a verification nature.

Ι option

ΙΙ option

1. In the rectangle MPKH O is the intersection point of the diagonals, RA and BH are the perpendiculars drawn from the vertices P and H to the line MK. It is known that MA = OB. Find the angle ROM.
2. In the rhombus MPKH, the diagonals intersect at a point O. On the sides MK, KH, PH, points A, B, C are taken, respectively, AK = KV = PC. Prove that OA = OB and find the sum of the angles ROS and MOA.
3. Construct a square along a given diagonal so that two opposite vertices of this square lie on different sides of a given acute angle.

VI. Summing up the lesson. Evaluation.

• What types of symmetry did you get acquainted with in the lesson?
• What two points are said to be symmetrical about a given line?
• Which figure is said to be symmetrical with respect to a given line?
• What two points are said to be symmetrical with respect to the given point?
• Which figure is said to be symmetrical with respect to a given point?
• What is mirror symmetry?
• Give examples of figures that have: a) axial symmetry; b) central symmetry; c) both axial and central symmetry.
• Give examples of symmetry in animate and inanimate nature.

VII. Homework.

1. Individual: complete by applying axial symmetry (see fig. 7).

Rice. 7

2. Construct a figure symmetrical to the given one with respect to: a) a point; b) straight line (see Fig. 8, 9).

Rice. eight	Rice. nine

3. Creative task: "In the world of animals." Draw a representative from the animal world and show the axis of symmetry.

VIII. Reflection.

• What did you like about the lesson?
• What material was the most interesting?
• What difficulties did you encounter while completing the task?
• What would you change during the lesson?