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Lesson type: combined.

Lesson objectives:

• Consider axial, central and mirror symmetries as properties of some geometric figures.
• Teach to construct symmetrical points and recognize figures with axial symmetry and central symmetry.
• Improve problem solving skills.

Lesson objectives:

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

Lesson equipment:

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

During the classes

I. Organizational moment.

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

II. Introduction.

What is symmetry?

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

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

The word “symmetry” translated from Greek sounds like “harmony”, meaning beauty, proportionality, proportionality, uniformity in the arrangement of parts. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, and modern buildings.

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

III. Main part. Types of symmetry.

Central symmetry

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

Practical task.

1. Points are given A, IN And M M relative to the middle of the segment AB.
2. Which of the following letters have a center of symmetry: A, O, M, X, K?
3. Do they have a center of symmetry: a) a segment; b) beam; c) a pair of intersecting lines; d) square?

Axial symmetry

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

Practical task.

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

Mirror symmetry

Points A And IN are called symmetrical relative to the plane α (plane of symmetry) if the plane α passes through the middle of the segment AB and perpendicular to this segment. Each point of the α plane is considered symmetrical to itself.

Practical task.

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

Rice. 2

This is interesting.

Symmetry in living nature.

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

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

There is helical symmetry in the arrangement of leaves on plant stems. Positioned like a screw along the stem, the leaves seem to spread out in different directions and do not obscure each other from the light, although the leaves themselves also have an axis of symmetry. Considering overall plan structure of any animal, we usually notice a certain regularity in the arrangement of body parts or organs, which are repeated around a certain axis or occupy the same position in relation to a certain plane. This regularity is called body symmetry. The phenomena of symmetry are so widespread in the animal world that it is very difficult to indicate a group in which no symmetry of the body can be noticed. Both small insects and large animals have symmetry.

Symmetry in inanimate nature.

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

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

One cannot help but see symmetry in faceted gemstones. Many cutters try to give diamonds the shape of a tetrahedron, cube, octahedron or icosahedron. Since the garnet has the same elements as the cube, it is highly prized by gemstone connoisseurs. Art products of garnets were found in graves Ancient Egypt, dating back to the predynastic period (over two millennia BC) (see Fig. 5).

In the Hermitage collections special attention used gold jewelry of the ancient Scythians. Extraordinarily thin artwork golden wreaths, tiaras, wood and decorated with precious red-violet garnets.

One of the most obvious uses of the laws of symmetry in life is in architectural structures. This is what we see most often. In architecture, axes of symmetry are used as means of expression architectural design(see Fig. 6). In most cases, patterns on carpets, fabrics, and indoor wallpaper are symmetrical about the axis or center.

Another example of a person using symmetry in his practice is technology. In engineering, symmetry axes are most clearly designated where it is necessary to estimate the deviation from the zero position, for example, on the steering wheel of a truck or on the steering wheel of a ship. Or one of most important inventions of humanity, having a center of symmetry is the wheel, and the propeller and other technical means also have a center of symmetry.

"Look in the mirror!"

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

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

Rice. 4	Rice. 5	Rice. 6

IV. Physical education minute.

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

V. Independent testing work.

Ι option

ΙΙ option

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

VI. Summing up the lesson. Assessment.

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

VII. Homework.

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

Rice. 7

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

Rice. 8	Rice. 9

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

VIII. Reflection.

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

Goals:

• educational:
  • give an idea of ​​symmetry;
  • introduce the main types of symmetry on the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand ideas about famous figures, introducing properties associated with symmetry;
  • show the possibilities of using symmetry when solving various tasks;
  • consolidate acquired knowledge;
• general education:
  • teach yourself how to prepare yourself for work;
  • teach how to control yourself and your desk neighbor;
  • teach to evaluate yourself and your desk neighbor;
• developing:
  • intensify independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • develop a “shoulder sense” in students;
  • cultivate communication skills;
  • instill a culture of communication.

DURING THE CLASSES

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

Exercise 1(3 min).

- Let's take a sheet of paper, fold it into pieces and cut out some figure. Now let's unfold the sheet and look at the fold line.

Question: What function does this line serve?

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

– This means that 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 an 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 goes?

Suggested answer: Differently.

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

Suggested answer: A lot of.

– That’s right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)

Question: What other figures have more than one axis of symmetry?

Suggested answer: Square, rectangle, isosceles and equilateral triangles.

– Let’s consider volumetric figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry do the square, rectangle, equilateral triangle and the proposed three-dimensional figures have?

I distribute halves of plasticine figures to students.

Task 4 (3 min).

– Using the information received, complete the missing part of the figure.

Note: the figure can be both planar and three-dimensional. It is important that students determine how the axis of symmetry runs and complete the missing element. The correctness of the work is determined by the neighbor at the desk and evaluates how correctly the work was done.

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

Task 5 (group work 5 minutes).

– 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.

Elements of drawings are presented to students

Task 6 (2 minutes).

– Find the symmetrical parts of these drawings.

To consolidate the material covered, I suggest next tasks provided for 15 minutes:

Name them all equal elements triangle KOR and COM. What type of triangles are these?

2. Draw several isosceles triangles in your notebook with common ground equal to 6 cm.

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

– Our initial ideas about form date back to the 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 little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new stone Age, in the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making 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 does symmetry occur in nature?

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

– Symmetry can also be observed in architecture. When constructing buildings, builders strictly adhere to symmetry.

That's why the buildings turn out so beautiful. Also an example of symmetry is humans and animals.

Homework:

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

You will need

• - properties of symmetrical points;
• - properties of symmetrical figures;
• - ruler;
• - square;
• - compass;
• - pencil;
• - paper;
• - a computer with a graphics editor.

Instructions

Draw a straight line a, which will be the axis of symmetry. If its coordinates are not specified, draw it arbitrarily. On one side of this straight line place arbitrary point A. it is necessary to find a symmetrical point.

Helpful advice

Symmetry properties are used constantly in AutoCAD. To do this, use the Mirror option. For building isosceles triangle or isosceles trapezoid it is enough to draw the lower base and the angle between it and the side. Reflect them using the given command and extend sides to the required value. In the case of a triangle, this will be the point of their intersection, and for a trapezoid - set value.

You constantly encounter symmetry in graphic editors when you use the “flip vertically/horizontally” option. In this case, the axis of symmetry is taken to be a straight line corresponding to one of the vertical or horizontal sides of the picture frame.

Sources:

• how to draw central symmetry

Constructing a cross section of a cone is not so difficult task. The main thing is to follow a strict sequence of actions. Then this task will be easy to do and will not require much labor from you.

You will need

• - paper;
• - pen;
• - circle;
• - ruler.

Instructions

When answering this question, you must first decide what parameters define the section.
Let this be the straight line of intersection of the plane l with the plane and the point O, which is the intersection with its section.

The construction is illustrated in Fig. 1. The first step in constructing a section is through the center of the section of its diameter, extended to l perpendicular to this line. The result is point L. Next, draw a straight line LW through point O, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lie point Q, as well as the already shown point W. These are the first two points of the desired section.

Now draw a perpendicular MS at the base of the cone BB1 ​​and construct the generators perpendicular section O2B and O2B1. In this section, through point O, draw a straight line RG parallel to BB1. Т.R and Т.G are two more points of the desired section. If the cross section of the ball were known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical that has symmetry with respect to the segment QW. Therefore, you should build as many section points as possible in order to connect them later with a smooth curve to obtain the most reliable sketch.

Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and construct the corresponding guides O2A and O2N. Through t.O, draw a straight line passing through PQ and WG until it intersects with the newly constructed guides at points P and E. These are two more points of the desired section. Continuing in the same way, you can find as many points as you want.

True, the procedure for obtaining them can be slightly simplified using symmetry with respect to QW. To do this, you can draw straight lines SS’ in the plane of the desired section, parallel to RG until they intersect with the surface of the cone. The construction is completed by rounding the constructed polyline from chords. It is enough to construct half of the desired section due to the already mentioned symmetry with respect to QW.

Video on the topic

Tip 3: How to make a graph trigonometric function

You need to draw schedule trigonometric functions? Master the algorithm of actions using the example of constructing a sinusoid. To solve the problem, use the research method.

You will need

• - ruler;
• - pencil;
• - knowledge of the basics of trigonometry.

Instructions

Video on the topic

note

If the two semi-axes of a single-strip hyperboloid are equal, then the figure can be obtained by rotating a hyperbola with semi-axes, one of which is the above, and the other, different from the two equal ones, around the imaginary axis.

Helpful advice

When examining this figure relative to the Oxz and Oyz axes, it is clear that its main sections are hyperbolas. And when cutting this spatial figure rotation by the Oxy plane, its cross section is an ellipse. The neck ellipse of a single-strip hyperboloid passes through the origin of coordinates, because z=0.

The throat ellipse is described by the equation x²/a² +y²/b²=1, and the other ellipses are composed by the equation x²/a² +y²/b²=1+h²/c².

Sources:

• Ellipsoids, paraboloids, hyperboloids. Rectilinear generators

The shape of a five-pointed star has been widely used by man since ancient times. We consider its shape beautiful because we unconsciously recognize in it the relationships of the golden section, i.e. the beauty of the five-pointed star is justified mathematically. Euclid was the first to describe the construction of a five-pointed star in his Elements. Let's join in with his experience.

You will need

• ruler;
• pencil;
• compass;
• protractor.

Instructions

The construction of a star comes down to the construction and subsequent connection of its vertices to each other sequentially through one. In order to build the correct one, you need to divide the circle into five.
Build arbitrary circle using a compass. Mark its center with point O.

Mark point A and use a ruler to draw line segment OA. Now you need to divide the segment OA in half; to do this, from point A, draw an arc of radius OA until it intersects the circle at two points M and N. Construct the segment MN. The point E where MN intersects OA will bisect segment OA.

Restore the perpendicular OD to the radius OA and connect points D and E. Make a notch B on OA from point E with radius ED.

Now, using line segment DB, mark the circle by five equal parts. Label the vertices of the regular pentagon sequentially with numbers from 1 to 5. Connect the points in next sequence: 1 with 3, 2 with 4, 3 with 5, 4 with 1, 5 with 2. Here is the correct five-pointed star, in regular pentagon. This is exactly the way I built it

If you think for a minute and imagine any object in your mind, then in 99% of cases the figure that comes to mind will be correct form. Only 1% of people, or rather their imagination, will draw an intricate object that looks completely wrong or disproportionate. This is rather an exception to the rule and refers to unconventionally thinking individuals with a special view of things. But returning to the absolute majority, it is worth saying that a significant proportion the right items still prevails. In the article we'll talk exclusively about them, namely about the symmetrical drawing of them.

Drawing the right objects: just a few steps to the finished drawing

Before you start drawing symmetrical object, you need to select it. In our version it will be a vase, but even if it doesn’t in any way resemble what you decided to depict, don’t despair: all the steps are absolutely identical. Follow the sequence and everything will work out:

1. All objects of regular shape have a so-called central axis, which is definitely worth highlighting when drawing symmetrically. To do this, you can even use a ruler and draw a straight line down the center of the landscape sheet.
2. Next, look carefully at the item you have chosen and try to transfer its proportions onto a sheet of paper. This is not difficult to do if you mark light strokes on both sides of the line drawn in advance, which will later become the outlines of the object being drawn. In the case of a vase, it is necessary to highlight the neck, bottom and the widest part of the body.
3. Do not forget that symmetrical drawing does not tolerate inaccuracies, so if there are some doubts about the intended strokes, or you are not sure of the correctness of your own eye, double-check the laid down distances with a ruler.
4. The last step is connecting all the lines together.

Symmetrical drawing is available to computer users

Due to the fact that most of the objects around us have the correct proportions, in other words, they are symmetrical, the developers computer applications created programs in which you can easily draw absolutely everything. Just download them and enjoy creative process. However, remember, a machine will never be a substitute for a sharpened pencil and a sketchbook.

TRIANGLES.

§ 17. SYMMETRY RELATIVELY TO THE RIGHT STRAIGHT.

1. Figures that are 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 allowing the ink to dry, we bend the sheet of paper along this straight line so that one part of the sheet overlaps the other. This other part of the sheet will thus produce an imprint of this figure.

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

Two figures are called symmetrical with respect to a certain straight line if, when bending the drawing plane along this straight line, they are aligned.

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

From the definition of symmetrical figures it follows that all symmetrical figures are equal.

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

Suppose now we need to construct a segment C "D", symmetrical this segment CD relative to straight AB. Let's build points C" and D", symmetrical to the points C and D. If we bend the drawing plane along AB, then points C and D will coincide, respectively, with points C" and D" (Drawing 130). Therefore, the segments CD and C "D" will align, they will be symmetrical.

Let us now construct a figure symmetrical given polygon ABCDE relative to this axis of symmetry MN (Fig. 131).

To solve this problem, let’s drop the perpendiculars A A, IN b, WITH With, D d and E e to the axis of symmetry MN. Then, on the extensions of these perpendiculars, we plot the segments
A A" = A A, b B" = B b, With C" = Cs; d D"" =D d And e E" = E e.

The polygon A"B"C"D"E" will be symmetrical to the polygon ABCDE. Indeed, if you bend the drawing along a straight line MN, then the corresponding vertices of both polygons will align, and therefore the polygons themselves will align; this proves that the polygons ABCDE and A" B"C"D"E" are symmetrical about the straight line MN.

2. Figures consisting of symmetrical parts.

Often found geometric figures, which 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 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). The figures in drawings 134, a, b are exactly symmetrical.

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

It should be noted that symmetrical figures can be combined simply by moving along a plane only in some cases. To combine symmetrical figures, as a rule, it is necessary to turn one of them with the opposite side,