Presentation on the topic "central symmetry". Presentation for the lesson "Axial and Central Symmetry" Presentation on the topic of symmetry about a point

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Prepared by students of the X "A" class: Zatsepina Ekaterina, Pavlova Yulia.

central symmetry.

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Here are examples of figures with central symmetry: The simplest figures with central symmetry are the circle and the parallelogram. 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.

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Two points A and B are called symmetrical with respect to the point O if O is the midpoint of the segment AB. Point O is considered symmetrical to itself.

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For example: In the figure, the points M and M1, N and N1 are symmetrical about the point O, and the points P and Q are not symmetrical about this point.

M M1 N N1 R Q

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Central symmetry in a rectangular coordinate system:

If in a rectangular coordinate system point A has coordinates (x0; y0), then the coordinates (-x0; -y0) of point A1, symmetrical to point A with respect to the origin, are expressed by the formulas x0 = -x0 y0 = -y0

y x 0 A(x0;y0) A1(-x0;-y0) x0 -x0 y0 -y0

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The point O is the center of symmetry if, when rotated around the point O by 180 °, the figure passes into itself.

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The line also has central symmetry, but unlike other figures that have only one center of symmetry (point O in the figures), the line has an infinite number of them - any point on the line is its center of symmetry. An example of a figure that does not have a center of symmetry is a triangle.

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Application in practice: Examples of symmetry in plants:

The question of symmetry in plants arose as early as the 5th century BC. e. In ancient Greece, the Pythagoreans drew attention to the phenomenon of symmetry in wildlife in connection with the development of their doctrine of harmony. In the 19th century there were individual works relating to this topic. And in 1961, as a result of centuries of research dedicated to the search for the beauty and harmony of the nature around us, the science of biosymmetry appeared. Central symmetry is characteristic of various fruits: blueberries, blueberries, cherries, cranberries. Consider a section of any of these berries. In section, it is a circle, and the circle, as we know, has a center of symmetry. Central symmetry can be observed in the image of such flowers as a dandelion flower, a coltsfoot flower, a water lily flower, a chamomile core, and in some cases the image of the entire chamomile flower also has central symmetry. Its core is a circle, and therefore centrally symmetrical, since we know that a circle has a center of symmetry. The whole flower has central symmetry only in the case of an even number of petals. In the case of an odd number of petals, remember pansies, it has only an axial one. Conclusions: According to our observations, in any plant you can find some part of it that has axial or central symmetry. These can be leaves, flowers, stems, tree trunks, fruits, and smaller parts such as the flower core, pistil, stamens, and others. Axial symmetry is inherent in various types of plants and fungi, and their parts. Central symmetry is most characteristic of plant fruits and some flowers.

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Central symmetry in architecture:

In the second half of the 18th - the first third of the 19th century, St. Petersburg acquired the glorified A.S. Pushkin "strict, slender appearance", which gave the city the architecture of classicism. All buildings built in the style of classicism have clear rectilinear symmetrical compositions. At the beginning of the 19th century, according to the project of A.N. Voronikhin built an outstanding work of art - the Kazan Cathedral. In front of the Kazan Cathedral, monuments to M.I. Kutuzov and M.B. Barclay de Tolly, commanders who defeated Napoleon's army. An example modern buildings, built in the middle of the twentieth century, is the hotel "Pribaltiyskaya". Symmetry, as can be seen from the drawing, is present both in the overall composition and in each of its three components: the middle part is an arch with a dome and a peak at the top, two side wings of the hotel. Conclusions: The principles of symmetry are fundamental for any architect, but each architect solves the question of the relationship between symmetry and asymmetry in different ways. An asymmetrical building as a whole can be a harmonic composition of symmetrical elements. A successful solution is determined by the talent of the architect, his artistic taste and his understanding of beauty. Take a walk around our city and see that there can be a lot of successful solutions, but one thing remains unchanged - the architect's desire for harmony, and this is to some extent connected with symmetry.

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Hotel "Pribaltiyskaya"

Kazan Cathedral

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Central symmetry in zoology:

Consider how the animal world and symmetry are connected. Central symmetry is most characteristic of animals leading an underwater lifestyle. And there is also an example of asymmetric animals: ciliates-shoes and amoeba Conclusions: The symmetry of a living being is determined by the direction of its movement. For living beings, for which the leading direction is the direction of movement “forward”, axial symmetry is most characteristic. Since in this direction animals rush for food and in the same direction they are saved from their pursuers. And the violation of symmetry would lead to the deceleration of one of the sides and the transformation of the translational motion into a circular one. Central symmetry is more common in the form of underwater animals. Asymmetry can be observed in the example of the simplest animals.

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Central symmetry in transport:

Central symmetry is not compatible with the shape of ground and underground transport. The reason for this is its direction of movement. When considering the top view of a tram, electric locomotive, cart, we see that the axis of symmetry runs along the direction of movement. Thus, the central symmetry should be sought in air and underwater transport, i.e., in such forms where the directions: forward, backward, right, left, are equivalent. One such mode of transport is a hot air balloon. Another example of air transport is a parachute. Scientists attribute his invention to the 13th century. In our drawing, we presented a top view hot air balloon. Note that it is similar to the top view of a parachute. As we can see, this figure is centrally symmetrical. O is the center of symmetry. Further development parachute received in the invention of our scientists "inflatable braking device". It is intended for the descent of cargo and man from orbit. The inflatable braking device is an elastic shell that is inflated in space. It has a flexible thermal protection and an additional inflatable shell. On the basis of it, it is also planned to design rescue devices that can be used, for example, in case of a fire in multi-storey buildings. The top view of this device is a circle. And the circle, as we know, not only has axial symmetry, but also central. The center of symmetry coincides with the center of the circle. Conclusions: Top view and front view of various modes of transport has either central or axial symmetry. For the ground mode of transport, axial symmetry is more characteristic. The reason for this is the direction of its movement. Central symmetry is more common in the form of air and underwater transport, for which the directions: right, left, forward, backward are equivalent. The transport models of the future, to the same extent as the models of the present and the past, have various types.

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Inflatable braking device

train capsule

Parachute (top view)

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Axioms of stereometry and planimetry

Prepared by: student X "A" class Zatsepina Ekaterina.

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Axiom 1(C1): Whatever the plane, there are points that belong to this plane and points that do not.

A α , B α α Α in E

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Axiom 2(C2): If two different planes have a common point, then they intersect along one straight line passing through this point.

β A α A β ) α β = m U m

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Axiom 3(C3): If two different straight lines have a common point, then it is possible to draw a plane through them, and moreover, only one.

a b = d a, b, d α d a

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Topic "Axial symmetry"

Oleinikova Galina Mikhailovna,

Municipal State Educational Institution "Yablochno Secondary School"

Khokholsky municipal district Voronezh region

"Mathematics reveals order, symmetry and certainty, and these are the most important types of beauty."

Aristotle (384 - 322 BC)

Problem learning technology

Subject "Mathematics"

The purpose of the lesson: organization of productive activities of students aimed at achieving the following results:

metasubject results:

in cognitive activity:

to help students realize the social, practical and personal significance educational material;

use to understand the world various methods(observation, measurement, experience, experiment, simulation, etc.)

comparison, comparison, classification of objects and objects according to one or more proposed criteria;

independent performance of various creative works;

participation in project activities;

in information - communication activities:

creation of written statements that adequately convey what was heard and readinformation with a given degree of curtailment (briefly, selectively, full)

Bringing an exampleditch, selection of arguments, formulation of conclusions;

reflection in oraland writing the results of their activities;

at the ability to paraphrase a thought (explain "in other words");

use for solving cognitive and communicative problemsvarious sources of information, including encyclopedias, wordsri, Internet resources and other databases;

in reflective activity:

assessment of their educational achievements;

conscious definitionspheres of their interests and opportunities;

mastery of skills joint activities: matching and coordination activities with other participants; objective evaluation their contribution to the solution of the common tasks of the team;

evaluating one's activities in terms of moralnorms and aesthetic values;

compliance rules healthy lifestyle life.

personal results:

be able to confidently and easily perform geometric constructions;

be able to express their thoughts in writing;

be able to speak well and easily express their thoughts;

to form character;

learn to apply the acquired knowledge and skills to solving new problems;

reason logically;

be able to fix their own difficulties, identify their cause, build ways out of difficulties;

subject results :

be able to build points, figures, symmetrical data;

give examples of symmetrical objects of the reality around us;

conduct research on this topic in nature and architecture;

Mastering the methods of activity applicable in the lesson of mathematics with integration into anatomy, biology, ecology, culture of a healthy lifestyle, architecture.

Lesson type: study lesson.

Forms of work: individual, pair, group, frontal.

Equipment: computer room with Internet access, projector, screen, presentation, figurines-tokens, drawings, magnets, colored chalk; each student has a folder with a set of geometric models, school tools, colored paper, colored pencils, scissors.

Methods: explanatory and illustrative, partially exploratory, research, design.

Forms of cognitive activity of students: frontal, individual.

Previously, students from the first lesson of the topic "Axial Symmetry" are grouped (at will and interests) into 3 groups, equal in number, so that each group has students who have Internet access at home. Each group gets a mini-study task: symmetry in nature, human anatomy and architecture.

Groups are saved during the lesson. For each correct answer, the team receives a token. One figure - one point. The team with the most points gets a score of 5; the other two conduct in-group self-assessment.

Actualization.

We live in a rapidly changing high-tech, information society, and we don’t think about why some objects and phenomena around us evoke a sense of beauty, while others do not.

In summer, a ladybug. Autumn yellow leaves on trees or leaves that have fallen to the ground are very beautiful. And in winter? - Snowflakes.

We are walking down the street and suddenly we slow down when we see a proportional and beautiful building.

Many people pass by, and each of us will pay attention to one person and say: "This person is beautiful and harmonious."

This chain can be continued, but now we are talking about something united: about the beauty, harmony and proportionality of living and inanimate nature.

I invite (I ask a specially trained) student of this class to come. Children pay attention to a symmetrical hairstyle, earrings, a blouse, a shawl with a symmetrical pattern.

Today we have your classmate visiting us and she is called ...

- "Symmetry".

And today we will touch on a wonderful mathematical phenomenon - axial symmetry. (slide 1-3)

Let's write down the topic of the lesson "Axial symmetry" in the notebook.

Today in the lesson we will try to answer the following questions:

What is symmetry?

What is axial symmetry?

Learn to identify symmetrical shapes.

Let's repeat the construction of symmetrical points and geometric figures with respect to a straight line.

What role does symmetry play in everyday human life (in nature, architecture, in everyday life)?
- Is it possible, knowing about the secret of harmony, to make the world better and more beautiful?

The teacher and students write down the number Classwork, the topic of the lesson on the board and in the notebook.

Then he invites the students to choose from the personal goals (or personal results) proposed on the screen, for the achievement of which each of them will try to work as much as possible in this lesson. Students determine for themselves personal results (selecting from the list on the screen) that they will strive for in the lesson, and the number of the goal (in the margins) in the notebook.

Frontal conversation.

What is symmetry? (slide 4-8)

The word symmetry has long been used in the meaning of harmony and beauty.

Euclid, Pythagoras, Leonardo da Vinci, Kepler and many other major thinkers of mankind tried to comprehend the secret of harmony.

“Symmetry is an idea with the help of which man has been trying for centuries to explain and create order, beauty, perfection” G. Weil.

What can you say about the meaning of the words "symmetry" and "axis"?

Symmetry is the sameness, proportionality in the arrangement of parts of something on opposite sides of a point, line or plane.

An axis is a straight line (an imaginary line passing through a geometric figure, which has only its inherent properties).

What points are called symmetrical?

Definition of symmetrical points about a straight line:

"Two points A and B are said to be symmetrical with respect to a line p if this line passes through the midpoint of the segment AB connecting these points and is perpendicular to it."

Formulate an algorithm for constructing a point symmetrical to a given one with respect to some line.

Why it will not be possible to complete the task, which sounds like this: " Build a figure symmetrical to this"?

This task is incomplete, since it is not clear whether the symmetry is performed with respect to a point or a straight line. This means that in order to perform axial symmetry, it is necessary to know the axis of symmetry.

Fixing the material.

1). Construction of a figure symmetrical to this one (relay race in groups)

Written work in notebooks and on the board. (Slide 9-12)

Exercise 1. Construct a point symmetrical to the given one with respect to the line a .

Task 2. Construct a line symmetrical to the given one with respect to the line m.

Task 3. Construct a triangle symmetrical to the given one with respect to the line n .

Task 4. Draw a figure by hand, symmetrical to the given one with respect to the vertical axis (tree, bird, cat). (Slide 13)

The figures are drawn on sheets and attached to the board. Everyone goes to the board and makes one element of the image, symmetrical to one figure from those proposed to his team. The team that completes the task first wins. Evaluation is carried out according to the following criteria:

Correct execution of the construction;

aesthetic perception;

The participation of each member of the group.

Exercise 5 (oral work ). Is it true that the following numerical intervals are sym are metric with respect to the line m, perpendicular to the coordinate line and passing through the origin O:

a) a segment from 3 to 7 and a segment from -7 to -3;

b) a segment from 10 to 25 and an interval from -25 to -10;

c) open rays from 1 to infinity and from minus infinity to 1?

Answer: a) yes; b) no; c) yes.

Task 6. Research"Find the axes of symmetry of the geometric figure."

How to determine if a figure has an axis of symmetry? (Slide 14-18)

Bend her over.

Yes, indeed, if they are bent along the depicted straight line, then its left and right parts will coincide. Such figures are symmetrical with respect to a straight line, and this straight line is an axis of symmetry.

How many axes of symmetry can a figure have? On the desks you have geometric shapes. Your task is to independently determine how many axes of symmetry each figure has. Determine the most "symmetrical" and the most "unsymmetrical" figure.

Students find the axes of symmetry of such geometric shapes as an angle, an equilateral, isosceles and scalene triangle, a rectangle, a rhombus, a square, a trapezoid, a parallelogram, a circle, an irregular polygon.

Let's find out which geometric shapes have one axis of symmetry?

Angle, isosceles triangle, trapezoid.

Two axes of symmetry?

Rectangle, rhombus.

Are the diagonals of a rectangle the axes of symmetry and why?

They are not, because when the rectangle is bent diagonally, the triangles do not match.

Students bend the figure diagonally and show that the parts of the rectangle do not match, that is, the diagonal of the rectangle is not an axis of symmetry.

Three axes of symmetry?

Equilateral triangle.

Four axes of symmetry?

Square.

How many axes of symmetry does a circle have?

A bunch of. These are straight lines passing through the center of the circle.

So which the most "symmetrical" and the most "asymmetrical" figure?

The most “symmetrical” is a circle, and the “asymmetrical” ones are a scalene triangle, a parallelogram; a polygon whose sides are not equal.

Task 7 ( Orally) . Can you give examples of symmetrical objects in your home and outdoor environment? Do we have symmetry?

Task 8 (Research and "local history" work-10 points).

I propose to conduct mini-studies in pairs or small groups, followed by a discussion about the presence of symmetry in the external and internal structure of humans, animals, plants; in the architecture of buildings of the countries of the world, our city and school.

When preparing messages, students use the Internet.

Results of mini-studies represent the students in the class. Each group of students presents the results of research on the following topics:

Axial symmetry and nature.

Axial symmetry and man.

Axial symmetry in architecture.

Create their own product in writing and presentation.

Protection is assessed by:

The optimally chosen material

Laconic presentation, logical reasoning,

aesthetic perception,

application in human life.

-"Axial symmetry in nature."(Slide 19-22)

Careful observation shows that the basis of the beauty of many forms created by nature is symmetry. Leaves, flowers, fruits have pronounced symmetry.

Ecological studies are closely related to the plants and trees around us.

By the symmetry of the birch leaves, one can speak of a healthy ecological situation in the microdistrict. If birch leaves are not symmetrical, then the ecological situation is unfavorable, this indicates the presence of radiation or chemical pollution. We are examining birch leaves collected in the microdistrict of western Bataysk. Based on the handout, we conclude that the ecological situation in the microdistrict is favorable.

It pours small grains from the sky, flies around the lanterns in huge fluffy flakes, stands as a pillar in the moonlight with ice needles. It would seem, what nonsense! Just frozen water. ... but how many questions a person has when looking at snowflakes.

Snowflake - This is a group of crystals formed from more than two hundred ice particles.

Symmetry - this is the property of crystals to be combined with each other in various positions by rotations, parallel transfers, reflections.

Calculate the axes of symmetry for your snowflake model.

- "Axial symmetry and fauna". (Slide 23)

Students note the symmetry of the external structure of animals, give examples of symmetrical color, but argue that the internal structure of animals is not symmetrical.

- "Axial symmetry and man". (Slide 24-25)

The beauty of the human body is due to proportionality and symmetry. The structure of the internal organs is not symmetrical.However, the human figure can be asymmetrical. One such example is scoliosis, a curvature of the spine, acquired, among other things, by poor posture.

Scoliosis - a lateral curvature of the spine - often occurs between the ages of 5 and 16 years. Among five-year-olds, scoliosis affects approximately 5-10% of children, by the end of school, scoliosis is detected in almost half of adolescents.

One of the main reasons is the wrong posture during training sessions, due to which there is an uneven load on the spine and muscles. Why is scoliosis dangerous and what diseases can it lead to in the future?

Most of the organs of the human body are directly controlled from the spinal cord through the spinal nerves. Infringement of the roots of nerves extending from the spinal cord leads to disruption of the internal organs. Hippocrates also pointed out the existence of a connection between the state of the spine and the functioning of internal organs. Prevention of scoliosis is better than its cure.

At the first signs of scoliosis, it is necessary to consult a specialist, follow a regimen that eases the load on the spine, provide nutrition rich in vitamins and minerals (the spine is in dire need of trace elements such as calcium, zinc, copper), you need to do morning exercises and exercise therapy. It is important to learn how to sit correctly at the desk: the back of the head should be slightly raised and laid back slightly, and the chin slightly lowered. With this position of the head, the entire spine is straightened and the blood supply to the brain improves. Feet should be on the floor, and the angle at the knee joints should be approximately 90 degrees.

The spine is one of the most important parts of the human body. Thanks to him, we can walk, run, jump, squat. The beauty and charm of a person largely depend on the posture.

80% of Russian children suffer from various types of postural disorders - from flat feet to scoliosis. The formation of the curves of the spine ends at the age of 6-7 and is fixed by the age of 14-17. This means that it is at this age that it is important for a teenager to develop the correct posture and thereby lay a reliable foundation for health for many years to come.

Violation of posture is not a disease, but a condition that needs to be corrected. They say that before the age of 21, while the body grows, many diseases of the musculoskeletal system can be cured. I suggest that all participants of our lesson follow the correct posture.

- "Axial symmetry in the architecture of the buildings of the cities of the world, the city of Bataysk."(Slide 26-32)

Symmetry is best seen in architecture. In the minds of ancient Greek architects, symmetry became the personification of regularity, expediency, and beauty. Examples of such structures are the Pyramid of Cheops in Egypt, Notre Dame Cathedral and the Eiffel Tower in France, Big Ben in the UK, the Taj Mahal mosque in Turkey.

The architecture of Russian Orthodox churches and cathedrals testifies that since ancient times, architectsthey knew well the mathematical proportion and symmetry and used them in the construction of architectural structures of Russia: the Kremlin, the Cathedral of Christ the Savior in Moscow, the Kazan and St. Isaac's Cathedrals in St. Petersburg, the cathedrals of Pskov, Nizhny Novgorod and others.

We asked ourselves one more question: “Do modern architects have the secret of creating beauty?” Our hometown is of interest. For example, the symbol of the city of Bataysk, which is located in the Central Park, fell in love with many citizens, we explain its aesthetic perception by the symmetry of its arch. We see symmetry in administrative, residential buildings, buildings cultural leisure.

The appearance of the Holy Trinity Church - the main attraction of the city, according to the architectural canons of the construction of Russian cathedrals, is an example of symmetry and proportionality. Studying the "Oath of Generations" memorial and monuments, we found out that they are based on symmetry. The building of the railway station of our city is also a sample of a symmetrical building. Thus, most of the buildings that form the face of our city are harmonious and comply with the laws of beauty.

- "Axial symmetry and our schoolyard." (Slide 33)

Examining the size of the native school, we see that the facade of the building, the porch, the section of the school fence, small architectural forms, flower beds comply with the rules of symmetry. So general form school yard looks harmonious.

Reflection. (Slide 34-37)

- The presentation slides show examples of symmetrical and non-symmetrical objects of the world (3 slides). Students are invited to identify patterns of symmetrical and asymmetrical objects, analyze why?

Homework:

- creative tasks on the topic "Statements of great scientists about symmetry";

- mini-presentations, photo reports about the symmetry of the surrounding reality;

- create models with symmetry using colored paper, scissors, felt-tip pens;

Owncreative task.

conclusions. (Slide 38)

Axial symmetry is a mathematical concept.

Learned to identify symmetrical shapes.

We learned how to build symmetrical points and geometric shapes relative to a straight line.

Symmetry is harmony.

The great thinkers of mankind tried to comprehend the secret of harmony. Today at the lesson, we also plunged into unraveling this mystery. We found out that symmetry plays one of the main directions in a person's daily life: in household items, in architecture, in nature.Knowing about the secret of harmony, one of which is axial symmetry, you can make the world a better and more beautiful place.

Do you know the famous phrase: “Beauty will save the world?” It is hard not to agree with Fyodor Mikhailovich Dostoevsky. We all want to make our lives more harmonious and beautiful. Guys, what do you think, maybe we have found the secret of creating beauty?

Lesson results.

Was an answer given to the problematic situation of the lesson, what new things were learned in the lesson, what did they learn, what caused difficulties and were they resolved in the lesson?

Grades are posted in the journal and diaries of students. The team with the highest score and students from other groups with high personal results receive a grade of 5; runner-up team - score 4.

Presentation “Movement. Central Symmetry" is a visual aid for conducting a mathematics lesson on this topic. With the help of the manual, it is easier for the teacher to form the student's idea of ​​​​central symmetry, to teach how to apply knowledge about this concept in solving problems. During the presentation, a visual representation of central symmetry is given, the definition of the concept is given, the properties of symmetry are noted, an example of solving a problem is described, in which the obtained theoretical knowledge is used.

The concept of motion is one of the most important mathematical concepts. It is impossible to consider it without a visual representation. A presentation is the best way to present the educational material on a given topic in the most understandable and profitable way. The presentation contains illustrations that help to quickly form an idea of ​​​​central symmetry, animation that improves the visibility of the demonstration and ensures a consistent presentation of educational material. The manual can accompany the teacher's explanation, helping him to achieve learning goals and objectives faster, contributing to the improvement of learning efficiency.

The demonstration starts by introducing the concept of central symmetry in the plane. The figure shows the plane α, on which point O is marked, with respect to which symmetry is considered. From the point o in one direction, the segment AO is laid off, equal to which A 1 O is laid off in the opposite direction from the center of symmetry. The figure shows that the constructed segments lie on one straight line. On the second slide, the concept is considered in more detail using the example of a point. It is noted that the central symmetry is the process of mapping some point K to point K 1 and vice versa. The figure shows such a display.

Slide 3 introduces the definition of central symmetry as a display of space, characterized by the transition of each point of a geometric figure to a symmetrical one with respect to the selected center. The definition is illustrated by a figure, which shows an apple and the mapping of each of its points to the corresponding point, symmetrical with respect to some point on the plane. Thus, we obtain a symmetrical image of an apple on a plane with respect to a given point.

On slide 4 the concept of central symmetry is considered in coordinates. The figure shows a spatial rectangular coordinate system Оxyz. A point M(x;y;z) is marked in space. Relative to the origin, M is displayed symmetrically and passes into the corresponding M 1 (x 1 ;y 1 ;z 1 ). The property of central symmetry is demonstrated. It is noted that the arithmetic mean of the corresponding coordinates of these points M(x;y;z), M 1 (x 1 ;y 1 ;z 1 ) is equal to zero, i.e. (x+ x 1)/2=0; (y + y 1)/2=0; (z+z 1)/2=0. This is equivalent to x=-x 1 ; y=-y 1 ; z=-z 1 . It is also noted that these formulas will be true even if the point coincides with the origin. Next, we prove the equality of the distances that are between points symmetrically reflected about the center of symmetry - a certain point. For example, some points A (x 1; y 1; z 1) and B (x 2; y 2; z 2) are indicated. Regarding the center of symmetry, these points are mapped to some points with opposite coordinates A(-x 1 ;-y 1 ;-z 1 ) and B(-x 2 ;-y 2 ;-z 2 ). Knowing the coordinates of the points and the formula for finding the distances between them, we determine that AB \u003d √ (x 2 -x 1) 2 + (y 2 -y 1) 2 + (z 2 -z 1) 2), and for the displayed points A 1 B 1 \u003d √ (-x 2 + x 1) 2 + (-y 2 + y 1) 2 + (-z 2 + z 1) 2). Given the properties of squaring, we can note the validity of the equality AB=A 1 B 1 . The preservation of distances between points with central symmetry indicates that it is a movement.

The solution of the problem is described, in which the central symmetry with respect to O is considered. The figure shows a straight line on which the points M, A, B are distinguished, the center of symmetry O, a straight line parallel to the given one, on which the points M 1, A 1 and B 1 lie. Segment AB is mapped to segment A 1 B 1 , point M - to point M 1 . For this construction, the equality of distances is noted, which is due to the properties of the central symmetry: OA=OA 1 , ∠AOB=∠A 1 OB 1 , OB=OB 1 . The equality of the two sides, angles means that the corresponding triangles are equal ΔАОB=ΔА 1 OB 1 . It is also indicated that the angles ∠ABO \u003d ∠A 1 B 1 O as lying across the straight lines A 1 B 1 and AB, therefore the segments AB and A 1 B 1 are parallel to each other. Further, it is proved that a line with central symmetry is mapped into a parallel line. One more point M, belonging to the line AB, is considered. Since the angles ∠MOA=∠M 1 OA 1 formed during the construction are equal as vertical, and ∠MAO=∠M 1 A 1 O are equal as cross-lying, and according to the construction, the segments OA=OA 1, then the triangles ΔMAO=ΔM 1 A 1 A. It follows from this that the distance MO \u003d M 1 O is preserved.

Accordingly, one can note the transition of the point M to M 1 with central symmetry, and the transition of M 1 to the point M with central symmetry with respect to O. The straight line passes into a straight line with central symmetry. On the last slide, you can use a practical example to consider central symmetry, in which each point of the apple and all its lines are displayed symmetrically, obtaining an inverted image.

Presentation “Movement. Central Symmetry" can be used to improve the effectiveness of a traditional school mathematics lesson on this topic. Also, this material can be successfully used to improve the clarity of the teacher's explanation when distance learning. For students who have not mastered the topic well enough, the manual will help to get a clearer idea of ​​​​the subject being studied.

Contents Central symmetry Central symmetry Central symmetry Central symmetry Tasks Tasks Tasks Construction Construction Construction Central symmetry in the environment Central symmetry in the environment Central symmetry in the environment Central symmetry in the environment Conclusion Conclusion Conclusion

Tasks 1. Segment AB, perpendicular to line c, intersects it at point O so that AOOB. Are points A and B symmetrical about point O? 2. Do they have a center of symmetry: a) a segment; b) beam; c) a pair of intersecting lines; d) square? A B C O 3. Construct an angle symmetrical to the angle ABC about the center O. Test yourself

5. For each of the cases shown in the figure, construct points A 1 and B 1, symmetrical to points A and B with respect to point O. B A A B AB O O O O C MP 4. Construct lines on which lines a and b with central symmetry with center O. Check yourself Help

7. Construct an arbitrary triangle and its image with respect to the point of intersection of its heights. 8. The segments AB and A 1 B 1 are centrally symmetrical with respect to some center C. Use one ruler to construct the image of the point M with this symmetry. A B A1A1 B1B1 M 9. Find points on lines a and b that are symmetrical with respect to each other. a b O Check yourself Help

Conclusion Symmetry can be found almost anywhere if you know how to look for it. Many peoples from ancient times owned the idea of ​​symmetry in a broad sense - as balance and harmony. Human creativity in all its manifestations gravitates toward symmetry. Through symmetry, man has always tried, in the words of the German mathematician Hermann Weyl, "to comprehend and create order, beauty and perfection."