TRIANGLES.

§ 17. SYMMETRY RELATIVELY DIRECT.

1. Figures symmetrical to each other.

Let's draw some figure on a sheet of paper with ink, and with a pencil outside it - an arbitrary straight line. Then, without letting the ink dry, fold the sheet of paper along this straight line so that one part of the sheet overlaps the other. On this other part of the sheet, the imprint of this figure will thus be obtained.

If you then straighten the sheet of paper again, then there will be two figures on it, which are called symmetrical relative to this straight line (Fig. 128).

Two figures are called symmetrical with respect to some straight line if they are combined when the plane of the drawing is folded along this straight line.

The line with respect to which these figures are symmetrical is called their axis of symmetry.

It follows from the definition of symmetrical figures that all symmetrical figures are equal.

You can get symmetrical figures without using the bending of the plane, but with the help of a geometric construction. Let it be required to construct a point C ", symmetrical to a given point C with respect to the straight line AB. Let us drop the perpendicular from the point C
CD to the straight line AB and on its continuation we set aside the segment DC "= DC. If we bend the plane of the drawing along AB, then the point C will coincide with the point C": points C and C "are symmetrical (Fig. 129).

Suppose now it is required to construct a segment C "D" symmetrical to a given segment CD with respect to the straight line AB. Let's build points C "and D", symmetrical to points C and D. If we bend the plane of the drawing along AB, then points C and D will coincide with points C "and D" (Fig. 130), respectively. Therefore, the segments CD and C "D" will coincide , they will be symmetrical.

Let us now construct a figure symmetrical to a given polygon ABCD with respect to a given axis of symmetry MN (Fig. 131).

To solve this problem, we drop the perpendiculars A a, AT b, FROM With, D d and E e on the axis of symmetry MN. Then, on the extensions of these perpendiculars, we set aside the segments
a A" = A a, b B" = B b, With C" \u003d Cs; d D""=D d and e E" = E e.

The polygon A "B" C "D" E "will be symmetrical to the polygon ABCD. Indeed, if the drawing is bent along the straight line MN, then the corresponding vertices of both polygons will coincide, which means that the polygons themselves will also coincide; this proves that the polygons ABCD and A" B"C"D"E" are symmetrical with respect to the straight line MN.

2. Figures consisting of symmetrical parts.

Often there are geometric figures that are divided by some straight line into two symmetrical parts. Such figures are called symmetrical.

So, for example, an angle is a symmetrical figure, and the bisector of the angle is its axis of symmetry, since when it is bent along it, one part of the angle is combined with the other (Fig. 132).

In a circle, the axis of symmetry is its diameter, since when bending along it, one semicircle is combined with another (Fig. 133). In the same way, the figures in the drawings 134, a, b are symmetrical.

Symmetrical figures are often found in nature, construction, and jewelry. The images placed on the drawings 135 and 136 are symmetrical.

It should be noted that symmetrical figures can be combined by simple movement along the plane only in some cases. To combine symmetrical figures, as a rule, it is necessary to turn one of them upside down,

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: A lot of.

- 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: a cube, a pyramid, a cone, a 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, 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.

The purpose of the lesson:

• formation of the concept of "symmetrical points";
• teach children to build points that are symmetrical to data;
• learn to build segments symmetrical to data;
• consolidation of the past (formation of computational skills, dividing a multi-digit number into a single-digit one).

On the stand "to the lesson" cards:

1. Organizational moment

Greetings.

The teacher draws attention to the stand:

Children, we begin the lesson by planning our work.

Today at the lesson of mathematics we will take a trip to 3 kingdoms: the kingdom of arithmetic, algebra and geometry. Let's start the lesson with the most important thing for us today, with geometry. I will tell you a fairy tale, but "A fairy tale is a lie, but there is a hint in it - a lesson for good fellows."

": One philosopher named Buridan had a donkey. Once, leaving for a long time, the philosopher put two identical armfuls of hay in front of the donkey. He put a bench, and to the left of the bench and to the right of it at the same distance he put exactly the same armfuls of hay.

Figure 1 on the board:

The donkey walked from one armful of hay to another, but did not decide which armful to start with. And, in the end, he died of hunger.

Why didn't the donkey decide which handful of hay to start with?

What can you say about these armfuls of hay?

(The armfuls of hay are exactly the same, they were at the same distance from the bench, which means they are symmetrical).

2. Let's do some research.

Take a sheet of paper (each child has a sheet of colored paper on their desk), fold it in half. Pierce it with the leg of a compass. Expand.

What did you get? (2 symmetrical points).

How to make sure that they are really symmetrical? (fold the sheet, the points match)

3. On the desk:

Do you think these points are symmetrical? (No). Why? How can we be sure of this?

Figure 3:

Are these points A and B symmetrical?

How can we prove it?

(Measure distance from straight line to points)

We return to our pieces of colored paper.

Measure the distance from the fold line (axis of symmetry), first to one and then to another point (but first connect them with a segment).

What can you say about these distances?

(The same)

Find the midpoint of your segment.

Where is she?

(It is the point of intersection of the segment AB with the axis of symmetry)

4. Pay attention to the corners, formed as a result of the intersection of the segment AB with the axis of symmetry. (We find out with the help of a square, each child works at his workplace, one studies on the board).

Conclusion of children: segment AB is at right angles to the axis of symmetry.

Without knowing it, we have now discovered a mathematical rule:

If points A and B are symmetrical about a line or axis of symmetry, then the segment connecting these points is at a right angle, or perpendicular to this line. (The word "perpendicular" is written separately on the stand). The word "perpendicular" is pronounced aloud in unison.

5. Let's pay attention to how this rule is written in our textbook.

Textbook work.

Find symmetrical points about a straight line. Will points A and B be symmetrical about this line?

6. Working on new material.

Let's learn how to build points that are symmetrical to data about a straight line.

The teacher teaches to reason.

To construct a point symmetrical to point A, you need to move this point from the line by the same distance to the right.

7. We will learn to build segments that are symmetrical to data, relative to a straight line. Textbook work.

Students discuss at the blackboard.

8. Oral account.

On this we will finish our stay in the "Geometry" Kingdom and conduct a small mathematical warm-up, having visited the "Arithmetic" kingdom.

While everyone is working orally, two students work on individual boards.

A) Perform a division with a check:

B) After inserting the necessary numbers, solve the example and check:

Verbal counting.

1. The life expectancy of a birch is 250 years, and an oak is 4 times longer. How many years does an oak tree live?
2. A parrot lives on average 150 years, and an elephant is 3 times less. How many years does an elephant live?
3. The bear called guests to his place: a hedgehog, a fox and a squirrel. And as a gift they presented him with a mustard pot, a fork and a spoon. What did the hedgehog give the bear?

We can answer this question if we execute these programs.

• Mustard - 7
• Fork - 8
• Spoon - 6

(Hedgehog gave a spoon)

4) Calculate. Find another example.

• 810: 90
• 360: 60
• 420: 7
• 560: 80

5) Find a pattern and help write down the right number:

3	9	81
2		16

5	10	20
6		24

9. And now let's rest a little.

Listen to Beethoven's Moonlight Sonata. A moment of classical music. Students put their heads on the desk, close their eyes, listen to music.

10. Journey into the realm of algebra.

Guess the roots of the equation and check:

Students decide on the board and in notebooks. Explain how you figured it out.

11. "Blitz tournament" .

a) Asya bought 5 bagels for a rubles and 2 loaves for b rubles. How much does the whole purchase cost?

We check. We share opinions.

12. Summarizing.

So, we have completed our journey into the realm of mathematics.

What was the most important thing for you in the lesson?

Who liked our lesson?

I enjoyed working with you

Thank you for the lesson.

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 the human body (look at it for 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 marked and they are not perpendicular to the axis of symmetry, and not only the dimensions but also the angle of inclination will have to be 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 also has an 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.

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 understandable 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 rather complicated. 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

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

Biology

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

Botany

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

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

Zoology

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

Axial symmetry

	Different types of symmetry of physical phenomena: symmetry of electric and magnetic fields (Fig. 1) In mutually perpendicular planes, the propagation of electromagnetic waves is symmetrical (Fig. 2)	fig.1 fig.2
Art	Mirror symmetry can often be observed in works of art. 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 kind of symmetry. One of Raphael's best early works, The Betrothal of Mary, was created in 1504. A valley topped with a white-stone temple stretches out under the sunny blue sky. In the foreground is the betrothal ceremony. The High Priest brings the hands of Mary and Joseph closer together. Behind Mary is a group of girls, behind Joseph is a group of young men. Both parts of the symmetrical composition are held together by the oncoming movement of the characters. For modern tastes, the composition of such a picture is boring, because the symmetry is too obvious.

Chemistry	The water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the world of wildlife. It is a double-stranded high molecular weight polymer whose monomer is nucleotides. DNA molecules have a double helix structure built on the principle of complementarity.
architewho	Since ancient times, man has used symmetry in architecture. Ancient architects used symmetry especially brilliantly in architectural structures. Moreover, the ancient Greek architects were convinced that in their works they are guided by the laws that govern nature. Choosing symmetrical forms, the artist thereby expressed his understanding of natural harmony as stability and balance. The city of Oslo, the capital of Norway, has an expressive ensemble of nature and art. This is Frogner - park - a complex of landscape gardening sculpture, which was created over 40 years.	Pashkov House Louvre (Paris)

© Sukhacheva Elena Vladimirovna, 2008-2009