Draw a symmetric figure about direct. Symmetric drawing of items of the right form

Objectives:

• educational:
  • give an idea of \u200b\u200bsymmetry;
  • introduce the main types of symmetry on the plane and in space;
  • develop strong skills to build symmetric figures;
  • expand the ideas about famous figures, introducing the properties associated with symmetry;
  • show the use of symmetry when solving various tasks;
  • consolidate the knowledge gained;
• general educational:
  • teach to configure yourself to work;
  • to teach you to control the control and neighbor in the desk;
  • teach themselves to evaluate yourself and the neighbor on the desk;
• developing:
  • activate independent activity;
  • develop cognitive activity;
  • learn to generalize and systematize the information obtained;
• educational:
  • brought up students' feeling of shoulder ";
  • educate communicativeness;
  • we instill a culture of communication.

DURING THE CLASSES

Before each underlie scissors and sheet of paper.

Exercise 1(3 min).

- Take a sheet of paper, fold it to get it and cut some feature. Now we will send a sheet and look at the fold line.

Question: What function does this line perform?

Estimated answer: This line divides the figure in half.

Question: How are all the points of the figure on the two half-bodies?

Estimated answer: All points of halves are at an equal distance from the fold line and on 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 easy, it has a wonderful property (all points relative to it are at the same distance), this line is the axis of symmetry.

Task 2. (2 minutes).

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

Task 3. (5 minutes).

- Hold a circle in the notebook.

Question: Determine how the axis of symmetry passes?

Estimated answer: Differently.

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

Estimated answer: Lot.

- That's right, the circle has many axes of symmetry. The same wonderful figure is a ball (spatial figure)

Question: What other figures do not have one axis of symmetry?

Estimated answer: Square, rectangle, equilibrium and equilateral triangles.

- Consider volumetric figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Direct how many axes of symmetry at a square, rectangle, an equilateral triangle and the proposed volume figures?

I distribute student to half of plasticine figures.

Task 4. (3 min).

- Using the information obtained, pull the missing part of the figure.

Note: The figure may be plane, and volumetric. It is important that students determine how the axis of symmetry passes, and the missing element died. The correctness of the execution determines the neighbor in the desk, assesses how properly the work is done.

A line (closed, unlocked, with self-intersection, without self-intersection) is laid out of the lace on the desktop.

Task 5. (Group work 5 min).

- Determine the visual axis of symmetry and relative to it to complete the second part from the lace of another color.

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

The elements of the drawings are presented in front of students.

Task 6. (2 minutes).

- Find symmetrical parts of these drawings.

To secure the material passed, I propose the following tasks provided for 15 minutes:

Name all the equal elements of the triangle of the Cor and Com. What is the type of these triangles?

2. Increase in a notebook several equally chained triangles with a shared basis equal to 6 cm.

3. Design the segment AB. Build a direct perpendicular segment AV and passing through its middle. Mark on it points C and D so that the quadrilateral of the ASD has been symmetrical with respect to the direct AV.

- Our initial ideas about form belong to a very distant era of the ancient stone century - Paleolithic. During the hundreds of millennia of this period, people lived in caves, in conditions of little animal difference. People made tools for hunting and fisheries, developed a tongue to communicate with each other, and in the late Paleolithic era, decorated their existence, creating works of art, figurines and drawings in which a remarkable feeling of shape is found.
When there was a transition from simple collection of food to its active production, from hunting and fisheries to agriculture, humanity enters the new stone Age, in neolithic.
The man of Neolithic possessed a sharp sense of geometric shape. Firing and coloring of clay vessels, manufacture of reed mats, baskets, fabrics, later - the treatment of metals produced ideas about plane and spatial figures. Neolithic ornaments joined the eyes, detecting equality and symmetry.
- And where is symmetry occur in nature?

Estimated answer: Wings of butterflies, beetles, leaves of trees ...

- Symmetry can be observed in architecture. Building building, builders clearly adhere to symmetry.

Therefore, buildings are so beautiful. Also, an example of symmetry is a person, animals.

Task for the house:

1. Come up with your ornament, depict it on a sheet A4 sheet (can be drawn in the form of a carpet).
2. Draw butterflies, note where elements of symmetry are present.

Triangles.

§ 17. Symmetry relatively straight.

1. Figures, symmetrical to each other.

Draw on a sheet of paper ink some kind of figure, and the pencil outside it is an arbitrary straight line. Then, without giving the ink to dry, run the sheet of paper along this direct so that one part of the sheet leaf to another. On this other part of the sheet will turn out, thus, the imprint of this figure.

If then the sheet of paper is straightened again, then there are two figures that are called symmetric Regarding the direct (damn 128).

Two figures are called symmetrical relative to some straight, if, when the drawing plane is inflexion, they are combined.

Straight, with respect to which these figures are symmetrical, called them axis of symmetry.

From the definition of symmetric figures it follows that any symmetric figures are equal.

It is possible to obtain symmetric figures without using the transformation of the plane, but by geometric construction. Let it be necessary to build a point with ", symmetrical to this point with a relatively straight AB. Omit from point with perpendicular
CD on direct AB and on the continuation of it to postpone the segment DC "\u003d DC. If the plane of the drawing is running around the AV, then the point C is aligned with the point C": Points C and C "symmetrical (damn 129).

Let now be required to build a segment with "D", symmetrical to this segment CD relative to direct AV. We construct a point with "and D", symmetric points C and D. If the plane of the drawing on AB will overtake, then the points C and D are monitored, respectively, with points from "and d" (Damn 130). In this segment CD and C "D" are monitored They will be symmetrical.

We will now construct a figure, symmetric by this polygon AVDE relative to this axis of symmetry Mn (damn 131).

To solve this problem, we lower perpendicular a but, IN b., FROM from, D. d. and E. e. on the axis of symmetry Mn. Then we postpone the segments on the continuations of these perpendicular
butA "\u003d a but, b.In "\u003d in b., fromWith "\u003d ss; d.D "" \u003d D d. and e.E "\u003d E e..

The polygon A "in" C "D" E "will be a symmetric polygon ASDE. Indeed, if the drawing in a straight line Mn, the corresponding vertices of both polygons are monitored, and therefore the polygons themselves are monitored; this proves that the polygons of AVDE and A" In "With" D "E" symmetrical with respect to the direct Mn.

2. Figures consisting of symmetric parts.

Often there are geometric shapes that some straight are separated into two symmetric parts. Such figures are called symmetric.

For example, an angle is a symmetrical figure, and the bisector of the angle is its axis of symmetry, since when inhibiting it, one part of the angle is combined from the other (damn 132).

In the circle of the symmetry axis, it is its diameter, since when inhibiting on it, one semicircle is combined with another (damn 133). Similarly, symmetrical figures in the drawings 134, and, b.

Symmetric figures are often found in nature, construction, in jewelry. Images placed on drawings 135 and 136 are symmetrical.

It should be noted that symmetric figures combine simple movement along the plane can only be in some cases. To combine symmetric figures, as a rule, it is necessary to turn one of them to the opposite side,

• Central Symmetry
• Axial symmetry
• Conclusion

Definition

Symmetry (from Greek. Symmetria - proportionality), in a broad sense - the invariance of the structure of the material object relative to its transformations. Symmetry plays a huge role in art and architecture. But it can be noticed in music, and in poetry. Symmetry is widely found in nature, especially in crystals, in plants and animals. Symmetry can meet in other sections of mathematics, for example, when building graphs of functions.

Central Symmetry

Two points BUT and BUT 1 are called symmetrical about the point ABOUT , if a ABOUT - Mid-cut AA 1. Point ABOUT It is considered symmetric.

Building a point, centrally symmetric given

• Build a beam ao
• Measure the length of the cut of JSC
• Point A1 is symmetrical to the point A relative to the center of O.

BUT 1

Building a segment, centrally symmetric to this

• Build a beam ao
• Measure the length of the cut of JSC
• To postpone on the ray of JSC on the other side of the point of the segment OA 1, equal to the separation of OA.
• Build a beam in
• Measure the length of the segment in
• To postpone on the beam in the other side of the point of the segment OB 1, equal to the segment of the s.
• Connect points a 1 and in 1 segment

BUT 1

IN 1

BUT 1

FROM 1

IN 1

Central symmetric figures are equal

Building a figure, centrally symmetric given

Rotate point A. around 90 turn center °

BUT 1

90 °

Turns points to different angles

BUT 1

135 °

45 °

BUT 2

90 °

BUT 3

Axial symmetry

Transformation of Figure F. in figure F. 1, in which each of its dot goes to a point, symmetrical relative to this direct, is called symmetry conversion relatively direct but . Straight but called the axis of symmetry.

Construction of a point, symmetrical given

2. AO \u003d OA '

Building a segment of symmetric to this

• AA ' C, JSC \u003d OA'.
• BB ' C, in' \u003d O 'in'.

3. A 'B' - the desired segment.

Building a triangle symmetric to this

1. AA ' C AO \u003d OA'

2. BB ' C BO' \u003d O'B '

3. CC ' C C O "\u003d O" C'

4.  A'B 'C' is a desired triangle.

Building a figure, symmetrical given relative to the symmetry axis

Figures possessing one axis of symmetry

Angle

Isosceles

triangle

Equal trapezium

Figures possessing two axes of symmetry

Rectangle

Rhombus

Figures having more than two axes of symmetry

Square

Equilateral triangle

A circle

Figures not possessing axial symmetry

Arbitrary triangle

Parallelogram

Incorrect polygon

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

If for a minute to think and imagine a subject in my imagination, then in 99% of cases the figure that came to mind will proper form. Only 1% of people, more precisely their imagination, draws an intricate object that looks completely incorrectly or disproportionately. It is rather an exception to the rules and belongs to unconventional reflecting individuals with a special look at things. But returning to the absolute majority, it is worth saying that the essential share of the right objects is still dominated. The article will be discussed exclusively about them, namely about symmetric drawing.

Image of the right objects: just a few steps to the finished drawing

Before proceeding to drawing a symmetric subject, you need to choose it. In our version it will be a vase, but even if it does not remind you that you have decided to depict you, do not despair: all the steps are absolutely identical. Stick the sequence and everything will turn out:

1. All items of the right form have the so-called central axis, which, with symmetrical drawing, it is necessary to allocate. To do this, you can even use the line and spend the straight line in the center of the landscape.
2. Next, carefully look at your chosen item and try to transfer it proportions on a sheet of paper. This is easy if there is a lung touches on both sides of the line, which will subsequently become the outlines of the drawdered item. In the case of a vase, it is necessary to highlight the neck, bottomsheko and the widest part of the housing.
3. Do not forget that symmetrical drawing does not tolerate inaccuracies, so if there are some doubts about the outlined strokes, or you are not sure about the correctness of your own eyelash, check the pending distances using the line.
4. The last step is to connect all the lines together.

Symmetrical drawing available to computer users

Due to the fact that most of the subjects around us have right proportionsIn other words, symmetrical, computer applications developers have created programs that can easily draw absolutely everything. Just download them and enjoy the creative process. However, remember, the car will never be replaced by the sharp pencil and landscape sheet.

I. . Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, withmeasures)

Summarizing table (all properties, features)

II. . Applications of symmetry:

1) in mathematics

2) in chemistry

3) in biology, botany and zoology

4) in art, literature and architecture

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1. The basic concepts of symmetry and its types.

The concept of symmetry P. rit is over the entire history of mankind. It is already found at the origins of human knowledge. It originated in connection with the study of a living organism, namely a person. And used sculptors in the 5th century BC. e. The word "symmetry" Greek, it means "proportionality, proportionality, the same in the location of the parts." It is widely used without eliminating the direction of modern science. Many great people conceived about this pattern. For example, L. N. Tolstoy said: "Standing in front of a black board and draws different figures on it with chalk, I suddenly struck by thought: why the symmetry is understandable to the eye? What is symmetry? This congenital feeling, I answered myself. What is it based on? ". Really symmetry is pleasant to the eye. Who did not admire the symmetry of the creatures of nature: leaves, flowers, birds, animals; Or the creations of a person: buildings, technician, - all the fact that from childhood surrounds, by what seeks to beauty and harmony. Herman Vaile said: "Symmetry is the idea, through which a person has tried to comprehend and create order, beauty and perfection." Herman Vaile is a German mathematician. His activity falls on the first half of the twentieth century. It was he who formulated the definition of symmetry, established for what features to see the presence or, on the contrary, the absence of symmetry in one way or another. Thus, mathematically strict representation was formed relatively recently - at the beginning of the twentieth century. It is quite complicated. We turn out and once again recall those definitions that are given to us in the textbook.

2. Axial symmetry.

2.1 Main definitions

Definition. Two points a and a 1 are called symmetrical relatively direct A, if this direct passes through the middle of the segment AA 1 and perpendicular to it. Each point is straight and is considered symmetrical.

Definition. The figure is called symmetrical relatively direct butif for each figure of the figure symmetrical to her relative to the direct but also belongs to this figure. Straight but called the axis of symmetry shape. It is also said that the figure has axial symmetry.

2.2 Build plan

And so, to build a symmetric figure with a relatively straight line from each point, we carry out perpendicular to this direct and extend it to the same distance, mark the resulting point. So we do with each point, we get symmetric peaks of a new figure. Then they connect them successively and we obtain a symmetrical figure of this relative axis.

2.3 Examples of figures with axial symmetry.

3. Central Symmetry

3.1 Main definitions

Definition. Two points a and a 1 are called symmetrical relative to the point O, if the middle of the segment AA 1. The point o is considered symmetrical.

Definition. The figure is called symmetrical about the point O if for each figure of the figure symmetrical to it relative to the point about also belongs to this figure.

3.2 Construction plan

Building a triangle symmetric given relative to the center of O.

To build a point, symmetrical point BUTrelative to the point ABOUT, it is enough to spend a straight OA(Fig. 46 ) and on the other side of the point ABOUTsqueeze OA. In other words , points A I. ; In and ; With I. Symmetrical relative to some point O. in fig. 46 Built triangle, symmetrical triangle ABC relative to the point ABOUT.These triangles are equal.

Building symmetric points relative to the center.

In the figure of the point M and M 1, N and N 1, symmetrical with respect to the point O, and the points p and q are not symmetric about this point.

In general, figures, symmetrical relative to some point, are equal .

3.3 examples

We give examples of figures with central symmetry. The simplest figures possessing the central symmetry are circle and parallelograms.

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

Straight also has a central symmetry, however, in contrast to the circle and the parallelogram that have only one symmetry center (point OH in the figure), there are many infinitely many - any point direct is its center of symmetry.

The figures show the angle symmetrical relative to the vertex, the segment symmetric to another segment relative to the center BUT and quadrangle symmetric relative to its vertex M.

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

4. Outcome lesson

Summarizing the knowledge gained. Today, at the lesson, we met with two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.

Summarizing table

	Axial symmetry	Central Symmetry
Feature	All points of the figure should be symmetrical about some straight line.	All points of the figure should, symmetrical with respect to the point chosen as the symmetry center.
Properties	1. Symmetric dots lie on the perpendicular to the straight line. 3. Direct transition to straight, angles in equal angles. 4. Sizes and shapes are saved.	1. Symmetric dots lie on a straight line passing through the center and this point of the figure. 2. Distance from point to straight equal to the distance from a straight line to a symmetrical point. 3. Sizes and shapes are saved.

II. Application of symmetry

Mathematics	In the algebra lessons, we studied the graphs of the functions y \u003d x and y \u003d x The figures present various pictures depicted using parabola branches. (a) octahedron, (b) a rhombic dodecahedron, (c) hexagonal octahedron.
Russian language	Printed letters Russian alphabet also possess various types of symmetries. In Russian, there are "symmetric" words - palindromewhich can be read equally in two directions.	A d l m p t f w- vertical axis In e z to u u -horizontal axis Well n about x- and vertical and horizontal B and y r at tsch i - no axis Radar Shalash Alla Anna
Literature	There may be palindromic and suggestions. Brucers wrote a poem "Moon's voice", in which each line - Palindrome. Look at the quantities, A.S. Pushkin "Copper Horseman". If you hold the line after the second line, we can notice the elements of the axial symmetry	And the rose fell on the Lap of Azor. I am going with the Suddy's sword. (Derzhavin) "Search Taxi" "Argentina Manit Negra", "Appreciates Negro Argentine", "Lached on the shelf of Claop found." In granite dressed novel; Bridges hung over waters; Dark green gardens She was covered with islands ...

Biology

The human body is based on the principle of bilateral symmetry. Most of us consider the brain as a single structure, in reality it is divided into two halves. These two parts are two hemispheres - firmly adjacent to each other. In full accordance with the overall symmetry of the human body, each hemisphere represents an almost accurate mirror image of another The main movement of human body movements and its sensory functions is evenly distributed between two brain hemispheres. Left hemisphere controls the right side of the brain, and the right side is left.

Botany

The flower is considered symmetrical when each perianth consists of an equal number of parts. Flowers, having paired parts, are considered to be with double symmetry flowers, etc. Triple symmetry is common for single-bedroom plants, five - for dicotyledonous Characteristic feature The buildings of plants and their development are spirality. Pay attention to the shoots sent - this is also a peculiar view of the spiral - spiral. Still Goethe, who was not only a great poet, but also a naturalist, considered spirality one of characteristic signs All organisms, manifestation of the most intimate essence of life. Spearly twist the amplification of plants, the spirals are tissue growth in the trunks of trees, the spirals are located in sunflower, spiral movements are observed with the growth of roots and shoots.

A characteristic feature of the structure of plants and their development is spirality. Look at the pine bump. The scales on its surface are strictly natural - along two spirals that intersect approximately at right angles. The number of such spirals in pine cones is 8 and 13 or 13 and 21.

Zoology

Under the symmetry in animals, the correspondence in size, shape and outlines, as well as the relative location of the parts of the body, located on the opposite sides of the separating line. With radial or radiant symmetry, the body has a form of a short or long cylinder or a central axis vessel, from which part of the body is departed. These are intestinal, iglobler, starfish. With bilateral symmetry of the axes of symmetry, there are three, but symmetrical sides, only one pair. Because the other two sides are abdominal and dorsal - do not look like each other. This type of symmetry is characteristic of most animals, including insects, fish, amphibians, reptiles, birds, mammals.

Axial symmetry

	Different kinds Symmetry physical phenomena: Symmetry of electrical and magnetic fields (Fig. 1) In mutually perpendicular planes symmetrically distribution electromagnetic waves (Fig. 2)	fig.1 Fig.2
Art	In artwork, it is often possible to observe a mirror symmetry. Mirror "symmetry is widely found in the works of art of primitive civilizations and in ancient painting. Medieval religious paintings are also characterized by this type of symmetry. One of the best early works Raphael - "Mary's Break" - created in 1504. Under the sunny blue sky, a valley spread out, crowned with a white-stone temple. In the foreground - the rite of engagement. The high priest brings the hands of Mary and Joseph. For Maria - a group of girls, for Joseph - young men. Both parts of the symmetric composition are bonded by the oncoming movement of characters. On a modern taste, the composition of such a painting is boring, as symmetry is too obvious.

Chemistry	Water molecule has a symmetry plane (straight vertical line). Externally important role in the world of wildlife, DNA molecules (deoxyribonucleic acid) are played. This is a two-chain high molecular weight polymer, whose monomer is nucleotides. DNA molecules have a double spiral structure built on the principle of complementarity.
Architecturecura	A person has long been using symmetry in architecture. The symmetry in architectural structures ancient architecture was particularly brilliantly used. Moreover, the ancient Greek architects were convinced that in their works they are guided by the laws that manage nature. Choosing symmetrical forms, the artist thereby expressed his understanding of natural harmony as stability and equilibrium. In the city of Oslo, the capital of Norway, there is an expressive ensemble of nature and artistic works. This is Frogner - Park - a complex of the garden sculpture, which was created for 40 years.	Pashkov House Louvre (Paris)

© Elena Vladimirovna Sukhacheva, 2008-2009.