Presentation on axial and central symmetry. Presentation for the lesson "Axial and central symmetry". Topic "Axial symmetry"

Movements.Movements
Central
.
symmetry
Completed by an 11th grade student
Heinrich Julia
The teacher checked
mathematicians Yakovenko Elena
Alekseevna
5klass.net Definition
Proof
Application in life
Application in nature
Problem solution

Central symmetry

B
DEFINITION:
A
Transformation Translating
each point A of the figure to point A1,
symmetrical with respect to it
center O, called central
symmetry.
C
ABOUT
C1
A1
O – center of symmetry
(point is stationary)
B1

Central symmetry

M
Points M and M1
are called
symmetrical
relative to point A,
if A is the middle
MM1.
A – center
symmetry
A
M1

The figure is called
symmetrical
relatively
center of symmetry,
if for each
figure points
symmetrical to her
point also
belongs to this
figure.

However, it can be noted that

a special case of rotation, namely,
turn 180 degrees.
Indeed, let at central
symmetry about point O point
X went to X". Then angle XOX"=180
degrees, as expanded, and XO=OX",
therefore such a transformation
is a 180 degree rotation.
It also follows that
central symmetry is
movement.

We are aware of planimetry
got acquainted with the movements
planes, i.e.
mappings of the plane onto
themselves, preserving
distances between points.
Let us now introduce the concept
movement of space.
Let us first clarify,
what is meant by words
display of space on

Let us assume that each point M
space is placed in
correspondence some point
M1, and any point of M1
space turned out to be
harmonized
some point M. Then
they say it's given
display of space on
myself.

M
A
M1
Movement
space is a mapping
space on
myself,
preserving
distance
between points.

Central symmetry is
movement that changes directions
opposite. That is, if at
central symmetry about point O
points X and Y correspond to points X" and Y", then
XY= - X"Y"
Proof:
Since point O is the midpoint of segment XX", then
obviously,
OX"= - OX
Likewise
OY"= - OY
Taking this into account, we find the vector X"Y":
X"Y"=OY"OX"=OY+OX=(OYOX)=XY
Thus, X"Y"=XY.

The proven property is
characteristic property
central symmetry, and
exactly the opposite is true
statement which is
sign of central
symmetry: "Movement,
changing directions to
the opposite is
central symmetry."

Task:

Prove that for central
symmetry:
a) a straight line that does not pass through the center
symmetry, displayed on
a line parallel to it;
b) a straight line passing through the center
symmetry, maps onto itself.

Symmetry can be
found almost everywhere
if you know how to look for it.
Many peoples with
ancient times
had an idea about
symmetry in the wide
sense - as in
poise and
harmony. Creation
people in all their
manifestations gravitate towards
symmetry. Through
symmetry man always
tried, according to
German mathematician
Hermann Weyl, “to comprehend and
create order, beauty and
perfection".
Conclusion

Presentation “Movements. Central symmetry" is a visual aid for teaching a mathematics lesson on this topic. With the help of the manual, it is easier for the teacher to form a student’s understanding of central symmetry and teach him to apply knowledge about this concept when solving problems. During the presentation, a visual representation of central symmetry, a definition of the concept is given, the properties of symmetry are noted, and an example of solving a problem in which the acquired theoretical knowledge is used is described.

The concept of motion is one of the most important mathematical concepts. It is impossible to consider it without a visual representation. Presentation is the best way to present educational material on a given topic in the most understandable and advantageous way. The presentation contains illustrations that help to quickly form an idea of ​​​​central symmetry, animation that improves the clarity of the demonstration and ensures a consistent presentation of educational material. The manual can accompany the teacher's explanation, helping him to quickly achieve educational goals and objectives, helping to increase the effectiveness of teaching.

The demonstration begins by introducing the concept of central symmetry on a plane. The figure shows the plane α, on which point O is marked, relative to which symmetry is considered. From point o, a segment AO is laid off in one direction, equal to which A 1 O is laid off in the opposite direction from the center of symmetry. The figure shows that the constructed segments lie on the same straight line. The second slide examines the concept in more detail using a point as an example. It is noted that central symmetry is the process of mapping a certain point K to point K 1 and back. The figure shows such a display.

Slide 3 introduces the definition of central symmetry as a display of space, characterized by the transition of each point of a geometric figure to symmetrical relative to the selected center. The definition is illustrated by a drawing that shows an apple and the mapping of each of its points to the corresponding point, symmetrical with respect to some point on the plane. Thus, we obtain a symmetrical image of an apple on a plane relative to a given point.

On slide 4, the concept of central symmetry is discussed in coordinates. The figure shows the spatial rectangular coordinate system Oxyz. A point M(x;y;z) is marked in space. Relative to the origin of coordinates, M is symmetrically displayed and goes into the corresponding M 1 (x 1 ;y 1 ;z 1 ). The property of central symmetry is demonstrated. It is noted that the arithmetic mean of the corresponding coordinates of these points M(x;y;z), M 1 (x 1 ;y 1 ;z 1 ) is equal to zero, that is, (x+ x 1)/2=0; (y+ y 1)/2=0; (z+z 1)/2=0. This is equivalent to x=-x 1 ; y=-y 1 ; z=-z 1 . It is also noted that these formulas will be true even if the point coincides with the origin of coordinates. Next, we prove the equality of the distances between points symmetrically reflected relative to the center of symmetry - a certain point. For example, some points A(x 1 ;y 1 ;z 1 ) and B(x 2 ;y 2 ;z 2 ) are indicated. With respect to the center of symmetry, these points are mapped to some points with opposite coordinates A(-x 1 ;-y 1 ;-z 1 ) and B(-x 2 ;-y 2 ;-z 2 ). Knowing the coordinates of the points and the formula for finding the distances between them, we determine 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). Taking into account the properties of squaring, we can note the validity of the equality AB = A 1 B 1. The preservation of distances between points with central symmetry indicates that it is a movement.

The solution to the problem is described in which central symmetry with respect to O is considered. The figure shows a straight line on which the points M, A, B are highlighted, the center of symmetry O, a straight line parallel to this one, on which the points M 1, A 1 and B 1 lie. Segment AB is mapped to segment A 1 B 1, point M is mapped to point M 1. For this construction, equality of distances is noted, which is due to the properties of central symmetry: OA=OA 1, ∠AOB=∠A 1 OB 1, OB=OB 1. The equality of two sides and angles means that the corresponding triangles are equal ΔAOB=ΔA 1 OB 1. It is also indicated that the angles ∠ABO=∠A 1 B 1 O lie crosswise at the lines A 1 B 1 and AB, therefore the segments AB and A 1 B 1 are parallel to each other. It is further proven that a straight line with central symmetry is mapped into a parallel straight line. We consider one more point M, belonging to the straight line AB. Since the angles ∠MOA=∠M 1 OA 1 formed during the construction are equal as vertical, and ∠MAO=∠M 1 A 1 O are equal as lying crosswise, and according to the construction the segments OA=OA 1, then the triangles ΔМАО=ΔМ 1 A 1 O. From this it follows that the distance MO = M 1 O is preserved.

Accordingly, we can note the transition of point M to M 1 with central symmetry, and the transition of M 1 to point M with central symmetry relative to O. A straight line with central symmetry turns into a straight line. On the last slide, you can use a practical example to consider central symmetry, in which each point of the apple and all its lines are displayed symmetrically, resulting in an inverted image.

Presentation “Movements. Central symmetry" can be used to improve the effectiveness of a traditional school mathematics lesson on this topic. Also, this material can be successfully used to improve the clarity of a teacher’s explanation during distance learning. For students who have not mastered the topic well enough, the manual will help them gain a clearer understanding of the subject being studied.

Slide 2

A B O Central symmetry is a mapping of space onto itself, in which any point goes into a point symmetric to it, relative to the center O. Point O is called the center of symmetry of the figure. Two points A and B are said to be symmetrical with respect to point O if O is the midpoint of segment AB. Point O is considered symmetrical to itself. In the figure, points M and M1, N and N1 are symmetrical relative to point O, but points P and Q are not symmetrical relative to this point. M M1 N N1 O P Q

Slide 3

Theorem. Central symmetry is movement.

Proof: Let, under central symmetry with the center at point O, points X and Y be mapped onto X" and Y". Then, as is clear from the definition of central symmetry, OX" = -OX, OY" = -OY. At the same time, XY = OY - OX, X"Y" = OY" - OX" Therefore, we have: X"Y" = -OY + OX = -XY It follows that central symmetry is a movement that changes direction to the opposite and vice versa, movement that reverses direction is central symmetry. Y" Y X" X O Property of central symmetry: central symmetry transforms a straight line (plane) into itself or into a straight line (plane) parallel to it.

Slide 4

Central symmetry in a rectangular coordinate system.

If in a rectangular coordinate system point A has coordinates (x0;y0), then the coordinates (-x0;-y0) of point A1, symmetrical to point A relative to the origin, are expressed by the formulas: x0 = -x0y0 = -y0 y x 0 A(x0 ;y0) А1(-x0;-y0) x0 -x0 y0 -y0

Slide 5

Examples from life.

The simplest figures with central symmetry are the circle and parallelogram. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the intersection of its diagonals. Central symmetry is found in the form of air and underwater transport (balloon, parachute), architecture, technology, art and everyday life. Central symmetry is most characteristic of plant fruits and some flowers (blueberries, blueberries, cherries, coltsfoot flowers, water lilies), as well as for animals leading an underwater lifestyle (amoeba). Oh Oh

Slide 6

One of the most beautiful examples of central symmetry is the snowflake. Many geometric bodies have central symmetry. These include all regular polyhedra (with the exception of the tetrahedron), all regular prisms with an even number of lateral faces, and some bodies of revolution (ellipsoid, cylinder, hyperboloid, torus, ball). Cube Octahedron Icosahedron Dodecahedron Three different hyperboloids

Slide 7

Examples of problem solving.

Given: ABCD is a parallelogram, triangles ABM, BCK, CDP, DAH are correct Prove: KPHM is a parallelogram Solution: Consider central symmetry (rotation by 180 degrees) about point O. Let f be central symmetry. f(B) = D, f(A) = C, f(D) = B, f(C) = A. With central symmetry f, the triangle BCK (regular) will transform into the equal triangle DAH (regular), according to the properties of the axial symmetry (angles are preserved). Similarly, triangle AMB transforms into triangle CPD. f(M) = P, f(K) = H, hence KO = OH, MO = OP, according to the parallelogram criterion, KPHM is a parallelogram.

Slide 8

Given: angle ABC, point D Construct a segment with ends on the sides of a given angle, the middle of which would be at point D Solution: Construct a point B "symmetrical to point B. Let D be the center of symmetry, BD = DB". Let's draw a line A"B" parallel to line BC and a line B"C" parallel to line AB. Lines A"B" and B"C" are symmetrical to straight lines BC and AB, respectively, with respect to point D. This means that point A" is symmetrical with point C" with respect to point D. It follows that A"D = DC".

View all slides

Axial and central symmetry

“ Symmetry is the idea through which man throughout the centuries tried to comprehend and create order, beauty and perfection.” German mathematician G. Weil

Symmetry (means “proportionality”) - the property of geometric objects to be combined with themselves under certain transformations. Symmetry is understood as any regularity in the internal structure of the body or figure.

Symmetry about a point is central symmetry, and symmetry about a straight line - this is axial symmetry.

Symmetry about a point assumes that there is something on both sides of the point at equal distances, for example other points or the locus of points (straight lines, curved lines, geometric figures).

Symmetry relative to a straight line (axis of symmetry) assumes that along a 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.

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 axial line - 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 perpendicular (axis of symmetry) to the middle of a segment is equidistant from the ends of this segment.

If you connect symmetrical points (points of a geometric figure) with a straight line through a symmetry point, then the symmetrical points will lie at the ends of the straight line, and the symmetry point will be its middle. If you fix the symmetry point and rotate the straight line, then the symmetrical points will describe curves, each point of which will also be symmetrical to the point of the other curved line.

Symmetry in architecture

Man has long used symmetry in architecture. The ancient architects made especially brilliant use of symmetry in architectural structures. Moreover, the ancient Greek architects were convinced that in their works they were guided by the laws that govern nature. By choosing symmetrical forms, the artist thereby expressed his understanding of natural harmony as stability and balance. Temples dedicated to the gods should be like this: the gods are eternal, they do not care about human concerns. The most clear and balanced buildings are those with a symmetrical composition. Symmetry gives harmony and completeness to ancient temples, towers of medieval castles, and modern buildings.

Sphinx at Giza

Aswan Mosque in Egypt

Symmetry in art

Symmetry is used in such forms of art as literature, Russian language, music, ballet, and jewelry.

If you look closely at the printed letters M, P, T, Ш, V, E, Z, K, S, E, ZH, N, O, F, X, you can see that they are symmetrical. Moreover, for the first four, the axis of symmetry runs vertically, and for the next six, it runs horizontally, and the letters Zh, N, O, F, X each have two axes of symmetry.

Ornament

Ornament (from Latin ornamentum - decoration) is a pattern consisting of repeating, rhythmically ordered elements. It can be tape (it is called a border), mesh or rosette. An ornament inscribed in a circle or in a regular polygon is called a rosette. The mesh design fills the entire flat surface with a continuous pattern. The border is obtained by parallel translation along a straight line.

Mirror symmetry

Symmetry relative to a plane is called mirror symmetry in some sources. Examples of figures - mirror reflections of one another - can be the right and left hands of a person, right and left screws, parts of architectural forms.

Man instinctively strives for stability, convenience, and beauty. Therefore, he is drawn to objects that have more symmetries. Why is symmetry pleasing to the eye? Apparently because symmetry prevails in nature. From birth, a person gets used to bilaterally symmetrical people, insects, birds, fish, and animals.

Celestial symmetry

• Every winter, myriads of snow crystals fall to the ground. Their cold perfection and absolute symmetry are amazing. Even adults during a snowfall enthusiastically, as in childhood, raise their faces to the sky, catch large snowflakes and fascinatedly look at the crystals that have landed on their palms. Among the snowflakes there are “plates”, “pyramids”, “columns”, “needles”, “steles” and “bullets”, simple or complex “stars” with highly branched rays - they are also called dendrites.
• Glaciologists - scientists who study the shape, composition and structure of ice, claim that each snow crystal is unique. However, all snowflakes have one thing in common - they have hexagonal symmetry. Therefore, the “stars” always grow three, six or twelve rays. The rarest twelve-pointed “star” is born in thunderclouds.
• The first systematic studies of snow crystals were undertaken in the 1930s by Japanese physicist Ukihiro Nakaya. He identified 41 types of snowflakes and compiled the first classification. In addition, the scientist grew the first “artificial” snowflake and found that the size and shape of the resulting ice crystals depend on air temperature and humidity.

Palindromes

Symmetry can also be seen in whole words, such as “Cossack”, “hut” - they are read the same both from left to right and from right to left. But here are entire phrases with this property (if you do not take into account the spaces between words): “Look for a taxi”,

"Argentina beckons the Negro"

“The Argentine appreciates the black man,”

“Lesha found a bug on the shelf,”

“And in the Yenisei there is blue,”

"City of Roads"

“Don’t nod (Don’t nod).”

Such phrases and words are called palindromes.

Drawings made by students

Symmetry is one of the most fundamental and one of the most general patterns of the universe: inanimate, living nature and society. We encounter symmetry everywhere. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human knowledge; it is widely used by all areas of modern science without exception.

Symmetry is present everywhere: in the regularity of day and night, seasons, in the rhythmic construction of a poem, practically wherever there is some kind of orderliness and regularity.

There are many types of symmetry in both the plant and animal worlds, but with all the diversity of living organisms, the principle of symmetry always operates, and this fact once again emphasizes the harmony of our world.

Contents Central symmetry Central symmetry Central symmetry Central symmetry Tasks Tasks Tasks Construction Construction Construction Central symmetry in the surrounding world Central symmetry in the surrounding world Central symmetry in the surrounding world Central symmetry in the surrounding world Conclusion Conclusion Conclusion

Problems 1. Segment AB, perpendicular to line c, intersects it at point O so that AOOB. Are points A and B symmetrical with respect to point O? 2. Do they have a center of symmetry: a) a segment; b) beam; c) a pair of intersecting lines; d) square? A B C O 3. Construct an angle symmetrical to angle ABC relative to center O. Test yourself

5. For each of the cases presented in the figure, construct points A 1 and B 1, symmetrical to points A and B relative to point O. B A A B A B O O O O S MP 4. Construct lines onto which lines a and are mapped b with central symmetry with center O. Test yourself Help

7. Construct an arbitrary triangle and its image relative to the point of intersection of its heights. 8. The segments AB and A 1 B 1 are centrally symmetrical with respect to some center C. Using one ruler, construct an image of the point M with this symmetry. A B A1A1 B1B1 M 9. Find points on lines a and b that are symmetrical relative to each other. a b O Test yourself Help

Conclusion Symmetry can be found almost everywhere if you know how to look for it. Since ancient times, many peoples have had an idea of ​​symmetry in the broad sense - as balance and harmony. Human creativity in all its manifestations tends towards symmetry. Through symmetry, man has always tried, in the words of the German mathematician Hermann Weyl, “to comprehend and create order, beauty and perfection.”