Symmetry. Axes of symmetry
Definition. Symmetry (means "proportionality") is the property of geometric objects to be combined with themselves under certain transformations. Under symmetry understand any correctness in the internal structure of the body or figure.
Point symmetry is the central symmetry (Figure 23 below), and symmetry about a straight line is axial symmetry (Figure 24 below).
Point symmetry assumes that there is something on both sides of a point at equal distances, for example, other points or a locus of points (straight lines, curved lines, geometric shapes).
If you connect a straight line symmetric points (points of a geometric figure) through a point of symmetry, then the symmetrical points will lie at the ends of the line, and the point of symmetry will be its midpoint. If you fix a point of symmetry and rotate a straight line, then the symmetrical points will describe curves, each point of which will also be symmetrical to a point on another curved line.
Symmetry about a straight line(axis of symmetry) assumes that along the perpendicular drawn through each point of the axis of symmetry, two symmetrical points are located at the same distance from it. The same geometric figures can be located relative to the axis of symmetry (straight line) as relative to the point of symmetry.
An example is a sheet of a notebook that is folded in half if you draw a straight line (axis of symmetry) along the fold line. Each point of one half of the sheet will have a symmetrical point on the other half of the sheet if they are located at the same distance from the fold line perpendicular to the axis.
The line of axial symmetry, as in Figure 24, is vertical, and the horizontal edges of the sheet are perpendicular to it. That is, the axis of symmetry serves as a perpendicular to the midpoints of the horizontal lines bounding the sheet. Symmetrical points (R and F, C and D) are located at the same distance from the center line - the perpendicular to the lines connecting these points. Consequently, all points of the perpendicular (axis of symmetry) drawn through the middle of the segment are equidistant from its ends; or any point of the perpendicular (axis of symmetry) to the middle of a line segment equidistant from the ends of this segment.
6.7.3. Axial symmetry
Points A and A 1 are symmetric with respect to the line m, since the line m is perpendicular to the segment AA 1 and goes through the middle.
m- axis of symmetry.
Rectangle ABCD has two axes of symmetry: straight m and l.
If the drawing is bent in a straight line m or in a straight line l, then both parts of the drawing will match.
Square ABCD has four axes of symmetry: straight m, l, k and s.
If the square is bent along any of the straight lines: m, l, k or s, then both parts of the square will coincide.
A circle centered at point O and radius OA has an infinite number of axes of symmetry. These are straight lines: m, m 1, m 2, m 3 .
Exercise. Construct point A 1, symmetric to point A (-4; 2) relative to the Ox axis.
Construct point A 2, symmetric to point A (-4; 2) relative to the axis Oy.
Point A 1 (-4; -2) is symmetrical to point A (-4; 2) relative to the Ox axis, since the Ox axis is perpendicular to the segment AA 1 and passes through its middle.
For points symmetric about the Ox axis, the abscissas coincide, and the ordinates are opposite numbers.
Point A 2 (4; -2) is symmetric to point A (-4; 2) relative to the axis Oy, since the axis Oy is perpendicular to the segment AA 2 and passes through its middle.
For points symmetric about the Oy axis, the ordinates coincide, and the abscissas are opposite numbers.
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Central and axial symmetry
Central symmetry
Two points A and A 1 are called symmetric with respect to point O if O is the middle of the segment AA 1 (Fig. 1). Point O is considered symmetrical to itself.
Central symmetry example
A figure is called symmetric about point O if for each point of the figure the point symmetric to it about point O also belongs to this figure. Point O is called the center of symmetry of the figure. The figure is also said to have central symmetry.
Examples of figures with central symmetry are a circle and a parallelogram (Figure 2).
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 Fig. 2), the straight line has infinitely many of them - any point of the straight line is its center of symmetry.
Axial symmetry
Two points A and A 1 are called symmetric with respect to the straight line a if this straight line passes through the middle of the segment AA 1 and is perpendicular to it (Fig. 3). Each point of the straight line a is considered symmetrical to itself.
A figure is called symmetrical with respect to a straight line a if for each point of a figure a point symmetrical to it with respect to a straight line a also belongs to this figure. Line a is called the axis of symmetry of the figure.
Examples of such figures and their axes of symmetry are shown in Figure 4.
Note that for a circle, any straight line passing through its center is an axis of symmetry.
Comparison of symmetries
Central and axial symmetry
How many axes of symmetry does the figure shown in the figure have?
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Lesson "Axial and central symmetry"
Brief description of the document:
Symmetry is a rather interesting topic in geometry, since this very concept is very often found not only in the process of human life but also in nature.
The first part of the video presentation "Axial and Central Symmetry" defines the symmetry of two points relative to a straight line on a plane. The condition for their symmetry is the possibility of drawing a segment through them, through the middle of which a given straight line will pass. A prerequisite for such symmetry is the perpendicularity of the line and the line.
The next part of the video lesson gives an illustrative example of the definition, which is shown in the form of a drawing, where several pairs of points are symmetrical about a straight line, and any point on this straight line is symmetrical to itself.
After gaining an initial understanding of symmetry, students are offered a more complex definition of a figure that is symmetrical about a straight line. The definition is offered in the form of a text rule, and is also accompanied in parallel by the speaker's speech off-screen. Completing this part are examples of symmetrical and non-symmetrical figures, relative to a straight line. It is interesting that there are geometric shapes that have several axes of symmetry - all of them are clearly presented in the form of drawings, where the axes are highlighted in a separate color. It is possible to facilitate the understanding of the proposed material in this way - an object or figure is symmetrical if it exactly coincides when the two halves are folded about its axis.
In addition to axial symmetry, there is symmetry about one point. The next part of the video presentation is devoted to this concept. First, the definition of the symmetry of two points relative to the third is given, then an example is provided in the form of a figure, which shows a symmetrical and non-symmetric pair of points. This part of the lesson ends with examples of geometric shapes with or without a center of symmetry.
At the end of the lesson, students are invited to familiarize themselves with the most striking examples of symmetry that can be found in the world around them. Understanding and the ability to build symmetrical figures are simply necessary in the life of people who are engaged in a variety of professions. At its core, symmetry is the basis of the entire human civilization, since 9 out of 10 objects surrounding a person have one or another type of symmetry. Without symmetry, it would not have been possible to build many large architectural structures, it would not have been possible to achieve impressive industrial capacities, and so on. In nature, symmetry is also a very common phenomenon, and if it is almost impossible to meet it in inanimate objects, then the living world is literally teeming with it - almost all flora and fauna, with rare exceptions, has either axial or central symmetry.
The normal school curriculum is designed in such a way that any student who is admitted to the lesson can understand it. A video presentation facilitates this process several times, since it simultaneously affects several centers of information assimilation, provides material in several colors, thereby forcing students to concentrate students' attention on the most important during the lesson. Unlike the usual way of teaching in schools, when not every teacher has the ability or desire to answer students to clarifying questions, the video lesson can be easily rewound to the necessary place in order to listen to the announcer again and read the necessary information again, until it is fully understood. Given the simplicity of the presentation of the material, the video presentation can be used not only during school hours, but also at home, as an independent way of learning.
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Presentation “Movement. Axial symmetry "
Archived documents:
Document's name 8.
Description of the presentation for individual slides:
Central symmetry is one example of movement
Definition Axial symmetry with the a-axis is a mapping of space onto itself, in which any point K passes into a point K1 symmetric to it with respect to the a-axis
1) Оxyz - rectangular coordinate system Оz - axis of symmetry 2) М (x; y; z) and M1 (x1; y1; z1), are symmetric about the axis Оz Formulas will also be true if the point М ⊂ Оz Axial symmetry is motion ZXY М (x; y; z) M1 (x1; y1; z1) O
Prove: Problem 1 with axial symmetry, a straight line forming an angle φ with the axis of symmetry is mapped onto a straight line that also forms an angle φ with the axis of symmetry. axis of symmetry angle φ AFEN mla φ φ
Given: 2) △ ABD - rectangular, according to the Pythagorean theorem: 1) DD1 ⏊ (A1C1D1), 3) △ BDD2 - rectangular, according to the Pythagorean theorem: Problem 2 Find: BD2 Solution:
Brief description of the document:
Presentation “Movement. Axial symmetry ”is a visual material for explaining the main provisions of this topic at a school mathematics lesson. In this presentation, axial symmetry is considered as another type of motion. During the presentation, the students are reminded of the studied concept of central symmetry, a definition of axial symmetry is given, the proposition that axial symmetry is motion is proved, and the solution of two problems in which it is necessary to operate with the concept of axial symmetry is described.
Axial symmetry is movement and is therefore difficult to represent on a chalkboard. Clearer understandable constructions can be made using electronic means. Thanks to this, constructions are clearly visible from any desk in the classroom. In the figures, it is possible to highlight the construction details in color, to focus on the features of the operation. Animation effects are used for the same purpose. With the help of presentation tools, it is easier for the teacher to achieve the learning goals, so the presentation is used to increase the effectiveness of the lesson.
The demonstration begins by reminding the students of the learned form of movement - central symmetry. An example of using the operation is the symmetrical display of a drawn pear. A point is marked on the plane, relative to which each point of the image turns into a symmetric one. The displayed image is thus inverted. In this case, all the distances between the points of the object are preserved with the central symmetry.
The second slide introduces the concept of axial symmetry. The figure shows a triangle, each of its vertices goes into a symmetrical vertex of the triangle about some axis. The definition of axial symmetry is highlighted in the box. It is noted that with it, each point of the object turns into a symmetric one.
Further, in a rectangular coordinate system, axial symmetry is considered, the properties of the coordinates of an object displayed using axial symmetry, and it is also proved that with this mapping, distances are preserved, which is a sign of motion. On the right side of the slide, the rectangular coordinate system Oxyz is depicted. The axis Oz is taken as the axis of symmetry. A point M is marked in space, which, under the corresponding mapping, is mapped to M 1. The figure shows that with axial symmetry, the point retains its applicate.
It is noted that the arithmetic mean of the abscissas and ordinates of the given mapping with axial symmetry is equal to zero, that is, (x + x 1) / 2 = 0; (y + y 1) / 2 = 0. Otherwise, it indicates that x = -x 1; y = -y 1; z = z 1. The rule also applies if the M point is marked on the Oz axis itself.
To consider whether the distances between the points are preserved in the case of axial symmetry, the operation on points A and B is described. Mapping about the Oz axis, the described points go into A1 and B1. To determine the distance between points, we will use the formula in which the distance is calculated by coordinates. It is noted that AB = √ (x 2 -x 1) 2 + (y 2 -y 1) 2 + (z 2 -z 1) 2), and for the displayed points A 1 B 1 = √ (-x 2 + x 1) 2 + (- y 2 + y 1) 2 + (z 2 -z 1) 2). Given the properties of squaring, it can be noted that AB = A 1 B 1. This suggests that distances are maintained between points - the main sign of movement. Hence, axial symmetry is motion.
On slide 5, the solution to problem 1 is considered. In it, it is necessary to prove the statement that a straight line passing at an angle φ to the axis of symmetry forms the same angle φ with it. An image is given to the problem on which the axis of symmetry is drawn, as well as a straight line m forming an angle φ with the axis of symmetry, and its mapping relative to the axis is a straight line l. The proof of the statement begins with the construction of additional points. It is noted that the line m intersects the axis of symmetry at A. If we mark the point F ≠ A on this line and drop the perpendicular from it to the axis of symmetry, we get the intersection of the perpendicular with the axis of symmetry at point E. With axial symmetry, the segment FE passes into the segment NE. As a result of this construction, we got right-angled triangles ΔAEF and ΔAEN. These triangles are equal, since AE is their common leg, and FE = NE are equal in construction. Accordingly, the angle ∠EAN = ∠EAF. It follows from this that the displayed line also forms an angle φ with the axis of symmetry. The problem has been solved.
On the last slide, the solution to Problem 2 is considered, in which you need to give a cube ABCDA 1 B 1 C 1 D 1 with side a. It is known that after symmetry about the axis containing the edge B 1 D 1, point D goes into D 1. In the task you need to find BD 2. The construction is done to the task. The figure shows a cube, which shows that the axis of symmetry is the diagonal of the cube face B 1 D 1. The segment formed during the movement of point D is perpendicular to the plane of the face to which the axis of symmetry belongs. Since the distance between the points is maintained during movement, then DD 1 = D 1 D 2 = a, that is, the distance DD 2 = 2a. From the right-angled triangle ΔABD by the Pythagorean theorem it follows that BD = √ (AB 2 + AD 2) = a√2. From a right-angled triangle ΔВDD 2 follows by the Pythagorean theorem BD 2 = √ (DD 2 2 + ВD 2) = a√6. The problem has been solved.
Presentation “Movement. Axial symmetry ”is used to improve the effectiveness of the school mathematics lesson. Also, this method of visibility will help the teacher doing distance learning. The material can be offered for independent consideration by students who have not mastered the topic of the lesson well enough.
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Human life is filled with symmetry. It is convenient, beautiful, there is no need to invent new standards. But what is she really and is it so beautiful in nature, as is commonly believed?
Symmetry
Since ancient times, people have sought to organize the world around them. Therefore, something is considered beautiful, but something is not very. From an aesthetic point of view, gold and silver ratios are considered attractive, as well as, of course, symmetry. This term is of Greek origin and literally means "proportionality". Of course, we are talking not only about coincidence on this basis, but also on some others. In a general sense, symmetry is a property of an object when, as a result of certain formations, the result is equal to the initial data. This is found in both living 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, are distinguished. The use of symmetry is also interesting, because it is found not only in nature, but also in ornaments on fabrics, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, since it is extremely exciting.
Use of the term in other scientific fields
In the following, 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. For example, the classification depends on which science this term refers to. So, the division into types varies greatly, although some of the basic ones, perhaps, remain the same everywhere.
Classification
There are several main 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;
- translational;
- screw;
- fractal;
- etc.
In biology, all species are called somewhat differently, although in essence they can be the same. Subdivision into certain groups occurs based on 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 reference 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 the point inside a figure or crystal, at which lines converge, connecting in pairs all sides parallel to each other. Of course, it does not always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since it does not exist. By definition, it is obvious that the center of symmetry is that through which a figure can be reflected back onto 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 is, of course, imaginary, but it is this plane that divides the figure into two equal parts to each other. It can pass through one or more sides, be parallel to it, or it can divide them. Several planes can exist for the same figure. These elements are commonly referred to as P.
But perhaps the most common is what is called the "axis of symmetry." This common phenomenon can be seen both in geometry and in nature. And it is worthy of separate consideration.
Axles
Often an element with respect to which a figure can be called symmetrical is
a straight line or segment protrudes. 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 as you like: divide the sides or be parallel to them, and also intersect the corners or not. Symmetry axes 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-mentioned 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
Conventionally, it is 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 a triangle, this element for a 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. In addition to all regular polygons and a ball, some cones, as well as pyramids, parallelograms and some others, will have at least one axis of symmetry. Each case must be considered separately.
Examples in nature
In life it is called bilateral, it occurs most
often. Any person and many animals are an example of this. The axial is called radial and is much less common, as a rule, in the plant kingdom. 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 by 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, radial symmetry is observed in many flowers: chamomile, cornflowers, sunflowers, etc. There are a lot of examples, they are literally everywhere around.
Arrhythmia
This term, first of all, reminds the majority of medicine and cardiology, however, it initially has a slightly different meaning. In this case, the synonym will be "asymmetry", that is, the absence or violation of regularity in one form or another. It can be seen as an accident, and sometimes it can be a wonderful technique, 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 clear that the faces and bodies of humans and animals are also not completely symmetrical. There have even been studies that have judged the "right" faces as inanimate or simply unattractive. Still, the perception of symmetry and this phenomenon in itself is amazing and has not yet been fully studied, and therefore extremely interesting.
Today we will talk about a phenomenon that each of us has to constantly meet in life: about symmetry. What is symmetry?
Approximately we all understand the meaning of this term. The dictionary says: symmetry is proportionality and full correspondence of the arrangement of parts of something relative to a straight line or point. Symmetry is of two types: axial and radial. Let's consider axial 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, you cannot find absolute symmetry! The halves of the leaf copy each other far from perfect, the same applies to the human body (take a closer look); it is the same with other organisms! By the way, it should be added that any symmetrical body is symmetrical with respect to the viewer in only one position. It is worth, say, turning the sheet, or raising one hand, and what? - you can see for yourself.
People achieve true symmetry in the works of their labor (things) - clothes, cars ... In nature, it is characteristic of inorganic formations, for example, crystals.
But let's get down to practice. It's not worth starting with complex objects like people and animals, let's try as the first exercise in a new field to finish drawing a mirror half of the sheet.
How to draw a symmetrical object - lesson 1
We make sure that it turns out as similar as possible. For 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 some anchor points for the future symmetrical line. We proceed as follows: we draw several perpendiculars to the axis of symmetry - the midrib of the leaf with a pencil without pressing. Four or five is enough for now. 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 a ruler, do not rely on the eye too much. As a rule, we tend to reduce the drawing - it has been noticed from experience. We do not recommend measuring distances with your fingers: the error is too large.
We connect the resulting points with a pencil line:
Now we are meticulously looking - are the halves really the same. If everything is correct, we will circle it with a felt-tip pen, we will clarify our line:
The poplar leaf has been finished, now you can swing at the oak one.
How to draw a symmetrical shape - lesson 2
In this case, the difficulty lies in the fact that the veins are indicated 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 accurately observed. Well, we 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 let's fix the theme - draw a symmetrical lilac leaf.
He also has an interesting shape - heart-shaped and with ears at the base you will have to pant:
So they drew:
Take a look at the resulting work from a distance and see how accurately we managed to convey the required similarity. Here's a tip: 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 it correctly) and cut the leaf along the original line. Look at the figure itself and the cut paper.
Motion concept
Let us first examine such a concept as movement.
Definition 1
Mapping a plane is called plane movement if the mapping maintains distances.
There are several theorems related to this concept.
Theorem 2
The triangle, when moving, goes into an equal triangle.
Theorem 3
Any figure, when moving, passes into a figure equal to it.
Axial and central symmetry are examples of motion. Let's consider them in more detail.
Axial symmetry
Definition 2
Points $ A $ and $ A_1 $ are called symmetric with respect to the line $ a $ if this line is perpendicular to the segment $ (AA) _1 $ and passes through its center (Fig. 1).
Picture 1.
Consider axial symmetry using the example of a problem.
Example 1
Construct a symmetrical triangle for this triangle relative to any of its sides.
Solution.
Let us be given a triangle $ ABC $. We will construct its symmetry with respect to the $ BC $ side. The $ BC $ side under axial symmetry will transform into itself (follows from the definition). Point $ A $ will move to point $ A_1 $ as follows: $ (AA) _1 \ bot BC $, $ (AH = HA) _1 $. The $ ABC $ triangle will be transformed into the $ A_1BC $ triangle (Fig. 2).
Figure 2.
Definition 3
A figure is called symmetric with respect to the straight line $ a $ if each symmetric point of this figure is contained in the same figure (Fig. 3).
Figure 3.
Figure $ 3 $ shows a rectangle. It has axial symmetry about each of its diameters, as well as about two straight lines that pass through the centers of opposite sides of this rectangle.
Central symmetry
Definition 4
Points $ X $ and $ X_1 $ are called symmetric with respect to the point $ O $ if the point $ O $ is the center of the segment $ (XX) _1 $ (Fig. 4).
Figure 4.
Let's consider the central symmetry on the example of the problem.
Example 2
Construct a symmetrical triangle for a given triangle at any of its vertices.
Solution.
Let us be given a triangle $ ABC $. We will construct its symmetry with respect to the vertex $ A $. The vertex $ A $ under the central symmetry goes over into itself (follows from the definition). Point $ B $ will go to point $ B_1 $ as follows $ (BA = AB) _1 $, and point $ C $ will go to point $ C_1 $ as follows: $ (CA = AC) _1 $. The $ ABC $ triangle will go into the $ (AB) _1C_1 $ triangle (Fig. 5).
Figure 5.
Definition 5
A figure is symmetric about the point $ O $ if each symmetrical point of this figure is contained in the same figure (Fig. 6).
Figure 6.
Figure $ 6 $ shows a parallelogram. It has central symmetry about the intersection of its diagonals.
Sample task.
Example 3
Let us be given a segment $ AB $. Construct its symmetry with respect to the line $ l $ that does not intersect the given segment and with respect to the point $ C $ lying on the straight line $ l $.
Solution.
Let us schematically depict the condition of the problem.
Figure 7.
Let us first draw the axial symmetry with respect to the straight line $ l $. Since axial symmetry is motion, then by Theorem $ 1 $, the segment $ AB $ will be mapped onto the segment equal to it $ A "B" $. To construct it, we will do the following: draw the lines $ m \ and \ n $ through the points $ A \ and \ B $, perpendicular to the line $ l $. Let $ m \ cap l = X, \ n \ cap l = Y $. Then we draw the segments $ A "X = AX $ and $ B" Y = BY $.
Figure 8.
Let us now depict the central symmetry about the point $ C $. Since the central symmetry is motion, then by Theorem $ 1 $, the segment $ AB $ will be mapped onto the segment equal to it $ A "" B "" $. To construct it, we will do the following: draw lines $ AC \ and \ BC $. Then we draw the segments $ A ^ ("") C = AC $ and $ B ^ ("") C = BC $.
Figure 9.
Goals:
- educational:
- give an idea of symmetry;
- to acquaint with the basic types of symmetry on the plane and in space;
- develop strong skills in building symmetrical figures;
- expand the understanding of known figures, introducing the properties associated with symmetry;
- show the possibilities of using symmetry in solving various problems;
- consolidate the knowledge gained;
- general educational:
- teach yourself to set yourself up for work;
- teach to control yourself and your neighbor on your desk;
- teach how to evaluate yourself and your deskmate;
- developing:
- to intensify independent activity;
- develop cognitive activity;
- teach to generalize and systematize the information received;
- educational:
- to instill in students a "sense of the shoulder";
- educate communication;
- instill a culture of communication.
DURING THE CLASSES
In front of each 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 figurine. Now expand the sheet and look at the fold line.
Question: What is the function of this line?
Supposed answer: This line divides the shape in half.
Question: How are all the points of the figure located on the two resulting halves?
Supposed answer: All points of the halves are at the same 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 are at the same distance relative to it), this line is the axis of symmetry.
Assignment 2 (2 minutes).
- Cut out a snowflake, find the axis of symmetry, characterize it.
Assignment 3 (5 minutes).
- Draw a circle in a notebook.
Question: Determine how the axis of symmetry runs?
Supposed answer: Differently.
Question: So how many axes of symmetry does a circle have?
Supposed answer: Lot.
- That's right, a circle has many axes of symmetry. The same remarkable figure is the ball (spatial figure)
Question: What other figures have more than one axis of symmetry?
Supposed answer: Square, rectangle, isosceles and equilateral triangles.
- Consider volumetric figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry a square, rectangle, equilateral triangle and the proposed volumetric figures have?
I distribute to students the halves of plasticine figures.
Assignment 4 (3 min).
- Using the information received, fill in the missing part of the figure.
Note: the figure can be both planar and volumetric. It is important that the students determine how the axis of symmetry goes and complete the missing piece. The correctness of the execution is determined by the neighbor on the desk, assesses how correctly the work has been done.
A line is laid out of a lace of the same color on the desktop (closed, open, with self-intersection, without self-intersection).
Assignment 5 (group work 5 min).
- Determine visually the axis of symmetry and build the second part from a lace of a different color relative to it.
The correctness of the work performed is determined by the students themselves.
The elements of the drawings are presented to the students
Assignment 6 (2 minutes).
Find the symmetrical parts of these patterns.
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 is the appearance of these triangles?
2. Draw in a notebook several isosceles triangles with a common base equal to 6 cm.
3. Draw line segment AB. Construct a straight line perpendicular to line segment AB and passing through its middle. Mark points C and D on it so that the quadrangle ACBD is symmetrical with respect to line AB.
- Our initial ideas about the form date back to a very distant era of the ancient Stone Age - the Paleolithic. For hundreds of millennia of this period, people lived in caves, in conditions that did not differ much from the life of animals. Humans made tools for hunting and fishing, developed languages to communicate with each other, and in the late Paleolithic era adorned their existence, creating works of art, figurines and drawings that 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, mankind enters a new Stone Age, the Neolithic.
Neolithic man had a keen sense of geometric shape. The burning and painting of earthen vessels, the making of reed mats, baskets, fabrics, and later - the processing of metals developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
- Where does symmetry occur in nature?
Supposed answer: wings of butterflies, beetles, tree leaves ...
“Symmetry can be seen in architecture as well. When constructing buildings, builders adhere to symmetry.
That is why the buildings are so beautiful. Also, an example of symmetry is a person, animals.
Home assignment:
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 the elements of symmetry are present.
