Mathematics teacher Kochkina L.K.

Subject AXIAL AND CENTRAL SYMMETRIES

Purpose of the lesson task:

To teach how to build symmetrical points and recognize figures with axial symmetry and central symmetry, the formation of students' spatial representations. Developing the ability to observe and reason; development of interest in the subject through the use of information technology. Development of mathematical competence of students. Raising a person who knows how to appreciate the beautiful.

Expected result Students will be able to build symmetrical figures about the center and the line.

Lesson equipment:

Use of information technologies (presentation).

During the classes

I. Organizational moment.

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

II. Presentation showing: "Symmetric World"(for students)

III. work on the topic of the lesson(group work)

Students complete assignments on their own. At the end, information is exchanged.

1 option

item 47

axial symmetry

Option 2

item 47

central symmetry

Not really

Not really

Consider the rules for constructing symmetrical figures.

1 .Central symmetry is symmetry about a point.

Points A and B are symmetrical with respect to some point O if the point O is the midpoint of the segment AB.

Algorithm for constructing a centrally symmetrical figure

We construct a triangle A 1 B 1 C 1, symmetrical to the triangle ABC, with respect to the center (point) O.

For this:

Connect the points A, B, C with the center O and continue these segments;

2. We measure the segments AO, VO, CO and set aside on the other side of the point O, segments equal to them (AO \u003d A 1 O 1, VO \u003d B 1 O 1, CO \u003d C 1 O 1);

3. Connect the resulting points with segments A 1 B 1, A 1 C 1, B 1 C 1.

4. Received ∆A 1 AT 1 With 1 symmetrical ∆ABC.

Point O is called the center of symmetry of the figure, and the figure is called centrally symmetrical.

Task number 1 The figure shows a part of the figure, the center of symmetry of which is the point M. Explain its construction

Task number 2 Check the correctness of the construction of the figure from No. 1 with a neighbor in the desk. Construct a quadrangle in his notebook and mark the point O, which does not belong to this quadrangle. Take your notebook back and construct a quadrilateral symmetrical to the given one with respect to point O.

Check the correctness of the completed task.

2. Axial symmetry - this is symmetry about the drawn axis (straight line).

Points A and B are symmetrical with respect to some line a if these points lie on a line perpendicular to the given one and at the same distance.

The axis of symmetry is called a straight line when bent along which the "halves" coincide, and the figure is called symmetrical about some axis.

Algorithm for constructing a figure symmetrical with respect to some straight line

We construct a triangle A 1 B 1 C 1 , symmetrical to the triangle ABC with respect to the line a.

For this:

1. We draw straight lines from the vertices of the triangle ABC perpendicular to the straight line a and continue them further.

2. We measure the distances from the vertices of the triangle to the resulting points on the straight line and plot the same distances on the other side of the straight line.

3. Connect the resulting points with segments A 1 B 1, B 1 C 1, B 1 C 1.

4. Received ∆ A 1 AT 1 With 1 symmetrical ∆ABC.

Tasks according to the textbook No. 248-252, No. 261

perform the construction of a figure symmetrical with respect to the straight line a (on the board and in notebooks).

VI. Summing up the lesson.

Reflection What types of symmetry did you meet in the lesson?

Homework:

Repeat definitions. Creative work: Having studied the Russian alphabet (for option 1) and the Latin alphabet (for option 2), choose those letters that have symmetry. To issue results of researches in the A4 format. Those who are interested in this topic can take part in the creative project "Symmetry in my favorite school"

Task number 4 Fill the table:

Line segment

Straight

Ray

Square

One center of symmetry

Infinitely many centers of symmetry

One axis of symmetry

Two axes of symmetry

Four axes of symmetry

Infinitely many axes of symmetry

1 option

item 47

axial symmetry

Option 2

item 47

central symmetry

Axial symmetry is symmetry about ____________

Central symmetry is symmetry about ________________

Two points A and A 1 are called symmetrical with respect to the line a if ____________

Two points A and A 1 are called symmetrical about the point O if _____________

The straight line a is called _______________

Point O is called _________________

A figure is called symmetrical with respect to a straight line, if for each point of the figure, the point symmetrical to it belongs to _________

A figure is called symmetrical with respect to the point O if for each point of the figure, the point symmetrical to it belongs to _______

Are figures that are symmetrical with respect to a straight line equal?

Not really

Are figures that are symmetrical about a point equal?

So, with regard to geometry: there are three main types of symmetry.

First of all, central symmetry (or symmetry about a point) - this is a transformation of the plane (or space), in which the only point (point O - the center of symmetry) remains in place, while the rest of the points change their position: instead of point A, we get point A1 such that point O is the middle of segment AA1. To construct a figure Ф1, symmetrical to the figure Ф with respect to the point O, it is necessary to draw a ray through each point of the figure Ф passing through the point O (the center of symmetry), and on this ray to set aside a point symmetrical to the one chosen with respect to the point O. The set of points constructed in this way will give a figure F1.

Of great interest are figures that have a center of symmetry: with symmetry about the point O, any point of the figure F is again transformed into some point of the figure F. There are many such figures in geometry. For example: a segment (the middle of the segment is the center of symmetry), a straight line (any of its points is the center of its symmetry), a circle (the center of the circle is the center of symmetry), a rectangle (the point of intersection of its diagonals is the center of symmetry). There are many centrally symmetric objects in animate and inanimate nature (student communication). Often people themselves create objects that have a center of symmetryrii (examples from needlework, examples from mechanical engineering, examples from architecture and many other examples).

Secondly, axial symmetry (or symmetry about a line) - this is a transformation of the plane (or space), in which only the points of the line p remain in place (this line is the axis of symmetry), while the rest of the points change their position: instead of the point B, we get such a point B1 that the line p is the perpendicular bisector to the segment BB1 . To construct a figure Φ1 symmetrical to the figure Φ with respect to the line p, it is necessary for each point of the figure Φ to construct a point symmetric to it with respect to the line p. The set of all these constructed points gives the required figure Ф1. There are many geometric shapes that have an axis of symmetry.

A rectangle has two, a square has four, a circle has any straight line passing through its center. If you look closely at the letters of the alphabet, then among them you can find those that have a horizontal or vertical, and sometimes both axes of symmetry. Objects with axes of symmetry are quite common in animate and inanimate nature (student reports). In his activity, a person creates many objects (for example, ornaments) that have several axes of symmetry.

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Thirdly, planar (mirror) symmetry (or symmetry about a plane) - this is a transformation of space, in which only points of one plane retain their location (α-plane of symmetry), the remaining points of space change their position: instead of point C, such a point C1 is obtained that the plane α passes through the middle of the segment CC1, perpendicular to it.

To construct a figure Ф1, symmetrical to the figure Ф with respect to the plane α, it is necessary for each point of the figure Ф to build points symmetrical with respect to α, they form the figure Ф1 in their set.

Most often, in the world of things and objects around us, we encounter three-dimensional bodies. And some of these bodies have planes of symmetry, sometimes even several. And the man himself in his activities (construction, needlework, modeling, ...) creates objects with planes of symmetry.

It is worth noting that along with the three listed types of symmetry, there are (in architecture)portable and swivel, which in geometry are compositions of several movements.

"Point of symmetry" - Symmetry in architecture. Examples of symmetry of plane figures. Two points A and A1 are called symmetric with respect to O if O is the midpoint of the segment AA1. Examples of figures with central symmetry are the circle and the parallelogram. Point C is called the center of symmetry. Symmetry in science and technology.

"Construction of geometric shapes" - Educational aspect. Control and correction of assimilation. The study of the theory on which the method is based. In stereometry - not strict constructions. Stereometric constructions. algebraic method. Transformation method (similarity, symmetry, parallel translation, etc.). For example: straight; angle bisector; median perpendicular.

"Human Figure" - The shape and movement of the human body is largely determined by the skeleton. Fair with a theatrical performance. Do you think there is a job for an artist in a circus? The skeleton plays the role of a frame in the structure of the figure. Main Body (belly, chest) Didn't pay attention Head, face, hands. A. Mathis. Proportions. Ancient Greece.

"Symmetry about a line" - Symmetry about a line is called axial symmetry. The straight line a is the axis of symmetry. Symmetry about a straight line. Bulavin Pavel, 9B class. How many axes of symmetry does each figure have? A figure can have one or more axes of symmetry. central symmetry. Equosceles trapezoid. Rectangle.

"Squares of figures geometry" - Pythagorean theorem. Areas of various figures. Solve the puzzle. Figures with equal areas are called equal areas. Area units. Area of ​​a triangle. Rectangle, triangle, parallelogram. square centimeter. Figures of equal area. Equal figures b). square millimeter. in). What will be the area of ​​the figure made up of figures A and D.

"Limit of a function at a point" - Then in this case. When striving. Limit of a function at a point. Continuous at a point. Equal to the value of the function in. But when calculating the limit of the function at. Equal to value. Expression. Aspiration. Or you can say this: in a sufficiently small neighborhood of the point. Compiled from. Decision. Continuous on intervals. In between.

Homothety and similarity.Homothety - a transformation in which each point M (plane or space) is assigned a point M", lying on OM (Fig. 5.16), and the ratio OM":OM= λ the same for all points other than O. fixed point O is called the homothety center. Attitude OM": OM considered positive if M" and M lie on one side of O, negative - on opposite sides. Number X is called the homothety coefficient. At X< 0 homothety is called inverse. Atλ = - 1 homothety becomes a symmetry transformation about a point O. With homothety, a straight line passes into a straight line, parallel lines and planes are preserved, angles (linear and dihedral) are preserved, each figure passes into it similar (Fig. 5.17).

The converse is also true. A homothety can be defined as an affine transformation in which the lines connecting the corresponding points pass through one point - the center of the homothety. Homothety is used to enlarge images (projection lamp, cinema).

Central and mirror symmetry.Symmetry (in a broad sense) is a property of a geometric figure Ф, characterizing a certain correctness of its form, its invariance under the action of movements and reflections. The figure Ф has symmetry (symmetric) if there are non-identical orthogonal transformations that take this figure into itself. The set of all orthogonal transformations that combine the figure Ф with itself is the group of this figure. So, a flat figure (Fig. 5.18) with a dot M, transforming-

Xia in yourself with a mirror reflection, symmetrical about the straight - axis AB. Here the symmetry group consists of two elements - the point M converted to M".

If the figure Ф on the plane is such that rotations about some point O through an angle of 360°/n, where n > 2 is an integer, transform it into itself, then the figure Ф has n-th order symmetry with respect to the point O - center of symmetry. An example of such figures is regular polygons, for example, star-shaped (Fig. 5.19), which has eighth order symmetry about its center. The symmetry group here is the so-called n-th order cyclic group. The circle has symmetry of infinite order (since it is combined with itself by turning through any angle).

The simplest types of spatial symmetry is central symmetry (inversion). In this case, with respect to the point O the figure Ф is combined with itself after successive reflections from three mutually perpendicular planes, i.e., the point O - the middle of the segment connecting the symmetrical points F. So, for the cube (Fig. 5.20) the point O is the center of symmetry. points M and M" cube

Human life is filled with symmetry. It is convenient, beautiful, no need to invent new standards. But what is she really and is she as beautiful in nature as is commonly believed?

Symmetry

Since ancient times, people have sought to streamline the world around them. Therefore, something is considered beautiful, and something not so. From an aesthetic point of view, golden and silver sections are considered attractive, as well as, of course, symmetry. This term is of Greek origin and literally means "proportion". Of course, we are talking not only about coincidence on this basis, but also on some others. In a general sense, symmetry is such a property of an object when, as a result of certain formations, the result is equal to the original data. It is found in both animate and inanimate nature, as well as in objects made by man.

First of all, the term "symmetry" is used in geometry, but finds application in many scientific fields, and its meaning remains generally unchanged. This phenomenon is quite common and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in ornaments on fabric, building borders and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely exciting.

Use of the term in other scientific fields

In the future, symmetry will be considered from the point of view of geometry, but it is worth mentioning that this word is used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon is studied from different angles and under different conditions. The classification, for example, depends on which science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged everywhere.

Classification

There are several basic types of symmetry, of which three are most common:

In addition, the following types are also distinguished in geometry, they are much less common, but no less curious:

• sliding;
• rotational;
• point;
• progressive;
• screw;
• fractal;
• etc.

In biology, all species are called somewhat differently, although in fact they can be the same. The division into certain groups occurs on the basis of the presence or absence, as well as the number of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

Some features are distinguished in the phenomenon, one of which is necessarily present. The so-called basic elements include planes, centers and axes of symmetry. It is in accordance with their presence, absence and quantity that the type is determined.

The center of symmetry is called the point inside the figure or crystal, at which the lines converge, connecting in pairs all sides parallel to each other. Of course, it doesn't always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since there is none. According to the definition, it is obvious that the center of symmetry is that through which the figure can be reflected to itself. An example is, for example, a circle and a point in its middle. This element is usually referred to as C.

The plane of symmetry, of course, is imaginary, but it is she who divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or it can divide them. For the same figure, several planes can exist at once. These elements are usually referred to as P.

But perhaps the most common is what is called "axes of symmetry." This frequent phenomenon can be seen both in geometry and in nature. And it deserves separate consideration.

axes

Often the element with respect to which the figure can be called symmetrical,

is a straight line or a segment. In any case, we are not talking about a point or a plane. Then the figures are considered. There can be a lot of them, and they can be located in any way: divide sides or be parallel to them, as well as cross corners or not. Axes of symmetry are usually denoted as L.

Examples are isosceles and In the first case there will be a vertical axis of symmetry, on both sides of which there are equal faces, and in the second the lines will intersect each angle and coincide with all bisectors, medians and heights. Ordinary triangles do not have it.

By the way, the totality of all the above elements in crystallography and stereometry is called the degree of symmetry. This indicator depends on the number of axes, planes and centers.

Examples in Geometry

It is conditionally possible to divide the entire set of objects of study of mathematicians into figures that have an axis of symmetry, and those that do not. All circles, ovals, as well as some special cases automatically fall into the first category, while the rest fall into the second group.

As in the case when it was said about the axis of symmetry of the triangle, this element for the quadrilateral does not always exist. For a square, rectangle, rhombus or parallelogram, it is, but for an irregular figure, accordingly, it is not. For a circle, the axis of symmetry is the set of straight lines that pass through its center.

In addition, it is interesting to consider volumetric figures from this point of view. At least one axis of symmetry, in addition to all regular polygons and the ball, will have some cones, as well as pyramids, parallelograms and some others. Each case must be considered separately.

Examples in nature

In life it is called bilateral, it occurs most
often. Any person and very many animals are an example of this. The axial one is called radial and is much less common, as a rule, in the plant world. And yet they are. For example, it is worth considering how many axes of symmetry a star has, and does it have them at all? Of course, we are talking about marine life, and not about the subject of study of astronomers. And the correct answer would be this: it depends on the number of rays of the star, for example, five, if it is five-pointed.

In addition, many flowers have radial symmetry: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.

Arrhythmia

This term, first of all, reminds most of medicine and cardiology, but it initially has a slightly different meaning. AT this case a synonym will be "asymmetry", that is, the absence or violation of regularity in one form or another. It can be found as an accident, and sometimes it can be a beautiful device, for example, in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous one is slightly inclined, and although it is not the only one, this is the most famous example. It is known that this happened by accident, but this has its own charm.

In addition, it is obvious that the faces and bodies of people and animals are also not completely symmetrical. There have even been studies, according to the results of which the "correct" faces were regarded as inanimate or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore extremely interesting.