The purpose of the lesson:
- formation of the concept of “symmetrical points”;
- teach children to construct points symmetrical to data;
- learn to construct segments symmetrical to data;
- consolidation of what has been learned (formation of computational skills, division of a multi-digit number by a single-digit number).
On the stand “for the lesson” there are cards:
1. Organizational moment
Greetings.
The teacher draws attention to the stand:
Children, let's start the lesson by planning our work.
Today in mathematics lesson we will take a journey into 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'll 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 placed a bench, and to the left of the bench and to the right of it, at the same distance, he placed completely identical armfuls of hay.
Figure 1 on the board:
The donkey walked from one armful of hay to another, but still did not decide which armful to start with. And, in the end, he died of hunger."
Why didn't the donkey decide which armful 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 a little 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 can you be sure they are truly symmetrical? (let's fold the sheet, the dots 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 this?
(Measure the distance from the straight line to the points)
Let's return to our pieces of colored paper.
Measure the distance from the fold line (axis of symmetry) first to one and then to the other point (but first connect them with a segment).
What can you say about these distances?
(The same)
Find the middle of your segment.
Where is it?
(Is the point of intersection of segment AB with the axis of symmetry)
4. Pay attention to the corners, formed as a result of the intersection of segment AB with the axis of symmetry. (We find out with the help of a square, each child works at his own workplace, one studies at the blackboard).
Children's conclusion: 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 straight line or axis of symmetry, then the segment connecting these points is at a right angle or perpendicular to this straight line. (The word “perpendicular” is written separately on the stand). We say the word “perpendicular” out loud in chorus.
5. Let us pay attention to how this rule is written in our textbook.
Work according to the textbook.
Find symmetrical points relative to the straight line. Will points A and B be symmetrical about this line?
6. Working on new material.
Let's learn how to construct points symmetrical to data relative to a straight line.
The teacher teaches reasoning.
To construct a point symmetrical to point A, you need to move this point from the straight line to the same distance to the right.
7. We will learn to construct segments symmetrical to data relative to a straight line. Work according to the textbook.
Students reason at the board.
8. Oral counting.
This is where we will end our stay in the “Geometry” Kingdom and will do a little mathematical warm-up by visiting the “Arithmetic” Kingdom.
While everyone is working orally, two students are working on individual boards.
A) Perform division with verification:
B) After inserting the required numbers, solve the example and check:
Verbal counting.
- The lifespan of a birch is 250 years, and an oak is 4 times longer. How long does an oak tree live?
- A parrot lives on average 150 years, and an elephant is 3 times less. How many years does an elephant live?
- The bear invited guests to him: 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
(The 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 required number:
3 9 81 2 16
5 10 20 6 24
9. Now let's rest a little.
Let's listen to Beethoven's Moonlight Sonata. A minute of classical music. Students put their heads on the desk, close their eyes, and listen to music.
10. Journey into the kingdom of algebra.
Guess the roots of the equation and check:
Students solve problems on the board and in notebooks. They explain how they guessed it.
11. "Blitz tournament" .
a) Asya bought 5 bagels for a rubles and 2 loaves for b rubles. How much does the entire purchase cost?
Let's check. Let's share our opinions.
12. Summarizing.
So, we have completed our journey into the kingdom of mathematics.
What was the most important thing for you in the lesson?
Who liked our lesson?
It was a pleasure working with you
Thank you for the lesson.
Let g be a fixed line (Fig. 191). Let's take an arbitrary point X and drop the perpendicular AX to the straight line g. On the continuation of the perpendicular beyond point A, we plot the segment AX" equal to the segment AX. Point X" is called symmetrical to point X relative to straight line g.
If a point X lies on a line g, then the point symmetrical to it is the point X itself. Obviously, the point symmetrical to the point X" is a point X.
The transformation of a figure F into a figure F", in which each of its points X goes to a point X", symmetrical with respect to a given straight line g, is called a symmetry transformation with respect to a straight line g. In this case, the figures F and F" are called symmetrical with respect to straight line g (Fig. 192).
If a symmetry transformation with respect to a line g takes a figure F into itself, then this figure is called symmetric with respect to a line g, and the line g is called the axis of symmetry of the figure.
For example, straight lines passing through the intersection point of the diagonals of a rectangle parallel to its sides are the axes of symmetry of the rectangle (Fig. 193). The straight lines on which the diagonals of a rhombus lie are its axes of symmetry (Fig. 194).
Theorem 9.3. The transformation of symmetry about a straight line is a movement.
Proof. Let us take this straight line as the y-axis of the Cartesian coordinate system (Fig. 195). Let an arbitrary point A (x; y) of the figure F go to the point A" (x"; y") of the figure F". From the definition of symmetry with respect to a straight line it follows that points A and A" have equal ordinates, and the abscissas differ only in sign:
x"= -x.
Let's take two arbitrary points A(x 1; y 1) and B (x 2; y 2) - They will go to points A" (- x 1, y 1) and B" (-x 2; y 2).
AB 2 = (x 2 - x 1) 2 + (y 2 - y 1) 2
A"B" 2 =(-x 2 + x 1) 2 +(y 2 -y 1) 2.
From this it is clear that AB = A "B". And this means that the transformation of symmetry about a straight line is motion. The theorem has been proven.
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 construct points C" and D", symmetrical to 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, segments CD and C "D" will coincide , 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,
In this lesson we will look at another characteristic of some figures - axial and central symmetry. We encounter axial symmetry every day when we look in the mirror. Central symmetry is very common in living nature. At the same time, figures that have symmetry have whole line properties. In addition, we subsequently learn that axial and central symmetries are types of movements with the help of which a whole class of problems is solved.
This lesson dedicated to axial and central symmetry.
Definition
The two points are called symmetrical relatively straight if:
In Fig. 1 shows examples of points symmetrical with respect to a straight line and , and .
Rice. 1
Let us also note the fact that any point on a line is symmetrical to itself relative to this line.
Figures can also be symmetrical relative to a straight line.
Let us formulate a strict definition.
Definition
The figure is called symmetrical relative to straight, if for each point of the figure the point symmetrical to it relative to this straight line also belongs to the figure. In this case the line is called axis of symmetry. The figure has axial symmetry.
Let's look at a few examples of figures that have axial symmetry and their axes of symmetry.
Example 1
The angle has axial symmetry. The axis of symmetry of the angle is the bisector. Indeed: let’s lower a perpendicular to the bisector from any point of the angle and extend it until it intersects with the other side of the angle (see Fig. 2).
Rice. 2
(since - the common side, 
(property of a bisector), and triangles are right-angled). Means, . Therefore, the points are symmetrical with respect to the bisector of the angle.
It follows from this that isosceles triangle has axial symmetry relative to the bisector (height, median) drawn to the base.
Example 2
An equilateral triangle has three axes of symmetry (bisectors/medians/altitudes of each of the three angles (see Fig. 3).
Rice. 3
Example 3
A rectangle has two axes of symmetry, each of which passes through the midpoints of its two opposite sides(see Fig. 4).
Rice. 4
Example 4
A rhombus also has two axes of symmetry: straight lines that contain its diagonals (see Fig. 5).
Rice. 5
Example 5
A square, which is both a rhombus and a rectangle, has 4 axes of symmetry (see Fig. 6).
Rice. 6
Example 6
For a circle, the axis of symmetry is any straight line passing through its center (that is, containing the diameter of the circle). Therefore, a circle has infinitely many axes of symmetry (see Fig. 7).
Rice. 7
Let us now consider the concept central symmetry.
Definition
The points are called symmetrical relative to the point if: - the middle of the segment.
Let's look at a few examples: in Fig. 8 shows the points and , as well as and , which are symmetrical with respect to the point , and the points and are not symmetrical with respect to this point.
Rice. 8
Some figures are symmetrical about a certain point. Let us formulate a strict definition.
Definition
The figure is called symmetrical about the point, if for any point of the figure the point symmetrical to it also belongs to this figure. The point is called center of symmetry, and the figure has central symmetry.
Let's look at examples of figures with central symmetry.
Example 7
For a circle, the center of symmetry is the center of the circle (this is easy to prove by recalling the properties of the diameter and radius of a circle) (see Fig. 9).
Rice. 9
Example 8
For a parallelogram, the center of symmetry is the point of intersection of the diagonals (see Fig. 10).
Rice. 10
Let's solve several problems on axial and central symmetry.
Task 1.
How many axes of symmetry does the segment have?
A segment has two axes of symmetry. The first of them is a line containing a segment (since any point on a line is symmetrical to itself relative to this line). The second is the perpendicular bisector to the segment, that is, a straight line perpendicular to the segment and passing through its middle.
Answer: 2 axes of symmetry.
Task 2.
How many axes of symmetry does a straight line have?
A straight line has infinitely many axes of symmetry. One of them is the line itself (since any point on the line is symmetrical to itself relative to this line). And also the axes of symmetry are any lines perpendicular to a given line.
Answer: there are infinitely many axes of symmetry.
Task 3.
How many axes of symmetry does the beam have?
The ray has one axis of symmetry, which coincides with the line containing the ray (since any point on the line is symmetrical to itself relative to this line).
Answer: one axis of symmetry.
Task 4.
Prove that the lines containing the diagonals of a rhombus are its axes of symmetry.
Proof:
Consider a rhombus. Let us prove, for example, that the straight line is its axis of symmetry. It is obvious that the points are symmetrical to themselves, since they lie on this line. In addition, the points and are symmetrical with respect to this line, since 
. Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the rhombus (see Fig. 11).
Rice. eleven
Draw a perpendicular to the line through the point and extend it until it intersects with . Consider triangles and . These triangles are right-angled (by construction), in addition, they have: - a common leg, and 
(since the diagonals of a rhombus are its bisectors). So these triangles are equal: 
. This means that all their corresponding elements are equal, therefore: . From the equality of these segments it follows that the points and are symmetrical with respect to the straight line. This means that it is the axis of symmetry of the rhombus. This fact can be proven similarly for the second diagonal.
Proven.
Task 5.
Prove that the point of intersection of the diagonals of a parallelogram is its center of symmetry.
Proof:
Consider a parallelogram. Let us prove that the point is its center of symmetry. It is obvious that the points and , and are pairwise symmetrical with respect to the point , since the diagonals of a parallelogram are divided in half by the point of intersection. Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the parallelogram (see Fig. 12).
symmetry architectural facade building
Symmetry is a concept that reflects the order existing in nature, proportionality and proportionality between the elements of any system or object of nature, orderliness, balance of the system, stability, i.e. some element of harmony.
Millennia passed before humanity, in the course of its social and production activities, realized the need to express in certain concepts the two tendencies it had established primarily in nature: the presence of strict orderliness, proportionality, balance and their violation. People have long paid attention to the correct shape of crystals, the geometric rigor of the structure of honeycombs, the sequence and repeatability of the arrangement of branches and leaves on trees, petals, flowers, and plant seeds, and reflected this orderliness in their practical activities, thinking and art.
Objects and phenomena of living nature have symmetry. It not only pleases the eye and inspires poets of all times and peoples, but allows living organisms to better adapt to their environment and simply survive.
In living nature, the vast majority of living organisms exhibit different kinds symmetries (shape, similarity, relative location). Moreover, organisms of different anatomical structure may have the same type of external symmetry.
The principle of symmetry states that if space is homogeneous, the transfer of a system as a whole in space does not change the properties of the system. If all directions in space are equivalent, then the principle of symmetry allows the rotation of the system as a whole in space. The principle of symmetry is respected if the origin of time is changed. In accordance with the principle, it is possible to make a transition to another reference system moving relative to this system at a constant speed. Inanimate world very symmetrical. Often symmetry violations in quantum physics elementary particles- this is a manifestation of an even deeper symmetry. Asymmetry is a structure-forming and creative principle of life. In living cells, functionally significant biomolecules are asymmetrical: proteins consist of levorotatory amino acids (L-form), and nucleic acids contain, in addition to heterocyclic bases, dextrorotatory carbohydrates - sugars (D-form), in addition, DNA itself is the basis of heredity is a right-handed double helix.
The principles of symmetry underlie the theory of relativity, quantum mechanics, physicists solid, nuclear and nuclear physics, particle physics. These principles are most clearly expressed in the invariance properties of the laws of nature. This is not only about physical laws, but also others, for example, biological. Example biological law conservation can serve as the law of inheritance. It is based on the invariance of biological properties with respect to the transition from one generation to another. It is quite obvious that without conservation laws (physical, biological and others), our world simply could not exist.
Thus, symmetry expresses the preservation of something despite some changes or the preservation of something despite a change. Symmetry presupposes the invariability not only of the object itself, but also of any of its properties in relation to transformations performed on the object. The immutability of certain objects can be observed in relation to various operations - rotations, translations, mutual replacement of parts, reflections, etc.
Let's consider the types of symmetry in mathematics:
- * central (relative to the point)
- * axial (relatively straight)
- * mirror (relative to the plane)
- 1. Central symmetry (Appendix 1)
A figure is said to be symmetrical with respect to point O if, for each point of the figure, a point symmetrical with respect to point O also belongs to this figure. Point O is called the center of symmetry of the figure.
The concept of a center of symmetry was first encountered in the 16th century. In one of Clavius’s theorems, which states: “if a parallelepiped is cut by a plane passing through the center, then it is split in half and, conversely, if a parallelepiped is cut in half, then the plane passes through the center.” Legendre, who first introduced the elements of the doctrine of symmetry into elementary geometry, shows that a right parallelepiped has 3 planes of symmetry perpendicular to the edges, and a cube has 9 planes of symmetry, of which 3 are perpendicular to the edges, and the other 6 pass through the diagonals of the faces.
Examples of figures that have central symmetry are the circle and parallelogram.
In algebra, when studying even and odd functions, their graphs are considered. When constructed, the graph of an even function is symmetrical with respect to the ordinate axis, and the graph of an odd function is symmetrical with respect to the origin, i.e. point O. So, not even function has central symmetry, and the even function is axial.
2. Axial symmetry (Appendix 2)
A figure is called symmetrical with respect to line a if, for each point of the figure, a point symmetrical with respect to line a also belongs to this figure. Straight line a is called the axis of symmetry of the figure. The figure is also said to have axial symmetry.
In a narrower sense, the axis of symmetry is called the axis of symmetry of the second order and speaks of “axial symmetry,” which can be defined as follows: a figure (or body) has axial symmetry about a certain axis if each of its points E corresponds to a point F belonging to the same figure, that the segment EF is perpendicular to the axis, intersects it and is divided in half at the intersection point.
I will give examples of figures that have axial symmetry. An undeveloped angle has one axis of symmetry - the straight line on which the angle's bisector is located. An isosceles (but not equilateral) triangle also has one axis of symmetry, and equilateral triangle-- three axes of symmetry. A rectangle and a rhombus, which are not squares, each have two axes of symmetry, and a square has four axes of symmetry. A circle has an infinite number of them - any straight line passing through its center is an axis of symmetry.
There are figures that do not have a single axis of symmetry. Such figures include a parallelogram, different from a rectangle, and a scalene triangle.
3. Mirror symmetry (Appendix 3)
Mirror symmetry (symmetry relative to a plane) is a mapping of space onto itself in which any point M goes into a point M1 that is symmetrical to it relative to this plane.
Mirror symmetry is well known to every person from everyday observation. As the name itself indicates, mirror symmetry connects any object and its reflection in a plane mirror. One figure (or body) is said to be mirror symmetrical to another if together they form a mirror symmetrical figure (or body).
Billiards players have long been familiar with the action of reflection. Their “mirrors” are the sides of the playing field, and the role of a ray of light is played by the trajectories of the balls. Having hit the side near the corner, the ball rolls towards the side located at a right angle, and, having been reflected from it, moves back parallel to the direction of the first impact.
It should be noted that two symmetrical figures or two symmetrical parts of one figure, despite all their similarities, equality of volumes and surface areas, in general case, are unequal, i.e. they cannot be combined with each other. These are different figures, they cannot be replaced with each other, for example, the right glove, boot, etc. not suitable for the left arm or leg. Items can have one, two, three, etc. planes of symmetry. For example, a straight pyramid, the base of which is an isosceles triangle, is symmetrical about one plane P. A prism with the same base has two planes of symmetry. A regular hexagonal prism has seven of them. Bodies of rotation: ball, torus, cylinder, cone, etc. have an infinite number of planes of symmetry.
The ancient Greeks believed that the universe was symmetrical simply because symmetry is beautiful. Based on considerations of symmetry, they made a number of guesses. Thus, Pythagoras (5th century BC), considering the sphere to be the most symmetrical and perfect form, concluded that the Earth is spherical and about its movement along the sphere. At the same time, he believed that the Earth moves along the sphere of a certain “central fire”. According to Pythagoras, the six planets known at that time, as well as the Moon, Sun, and stars, were supposed to revolve around the same “fire.”
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