Homothety and similarity.Homothety - a transformation in which each point M (plane or space) is assigned a point M", lying on OM (Fig. 5.16), and the ratio OM":OM= λ the same for all points other than O. fixed point O is called the homothety center. Attitude OM": OM considered positive if M" and M lie on one side of O, negative - on opposite sides. Number X is called the homothety coefficient. At X< 0 homothety is called inverse. Atλ = - 1 homothety becomes a symmetry transformation about a point O. With homothety, a straight line passes into a straight line, parallel lines and planes are preserved, angles (linear and dihedral) are preserved, each figure passes into it similar (Fig. 5.17).
The converse is also true. A homothety can be defined as an affine transformation in which the lines connecting the corresponding points pass through one point - the center of the homothety. Homothety is used to enlarge images (projection lamp, cinema).
Central and mirror symmetry.Symmetry (in a broad sense) is a property of a geometric figure Ф, characterizing a certain correctness of its form, its invariance under the action of movements and reflections. The figure Ф has symmetry (symmetric) if there are non-identical orthogonal transformations that take this figure into itself. The set of all orthogonal transformations that combine the figure Ф with itself is the group of this figure. So, a flat figure (Fig. 5.18) with a dot M, transforming-
Xia in yourself with a mirror reflection, symmetrical about the straight - axis AB. Here the symmetry group consists of two elements - the point M converted to M".
If the figure Ф on the plane is such that rotations about some point O through an angle of 360°/n, where n > 2 is an integer, transform it into itself, then the figure Ф has n-th order symmetry with respect to the point O - center of symmetry. An example of such figures is regular polygons, for example, star-shaped (Fig. 5.19), which has eighth order symmetry about its center. The symmetry group here is the so-called n-th order cyclic group. The circle has symmetry of infinite order (since it is combined with itself by turning through any angle).
The simplest types of spatial symmetry is central symmetry (inversion). In this case, with respect to the point O the figure Ф is combined with itself after successive reflections from three mutually perpendicular planes, i.e., the point O - the middle of the segment connecting the symmetrical points F. So, for the cube (Fig. 5.20) the point O is the center of symmetry. points M and M" cube
CHAPTER THREE
POLYHEDRALS
V. THE CONCEPT OF SYMMETRY OF SPATIAL FIGURES
99. Central symmetry. Two figures are called symmetrical with respect to any point O in space if each point A of one figure corresponds to a point A in the other figure, located on the straight line OA on the other side of the point O, at a distance equal to the distance of the point A from the point O (Fig. 114) Point O is called center of symmetry figures.
We have already seen an example of such symmetrical figures in space (§ 53), when, continuing beyond the vertex of the edges and faces of a polyhedral angle, we obtained a polyhedral angle symmetrical to the given one. The corresponding segments and angles that are part of two symmetrical figures are equal to each other. Nevertheless, the figures as a whole cannot be called equal: they cannot be combined with one another due to the fact that the order of the arrangement of parts in one figure is different than in another, as we saw in the example of symmetrical polyhedral angles.
In some cases, symmetrical figures can be combined, but at the same time their inconsistent parts will coincide. For example, let's take a right trihedral angle (Fig. 115) with a vertex at the point O and edges OX, OY, OZ.
Let's construct a symmetric angle OX"Y"Z" for it. Angle OXYZ can be combined with OX"Y"Z" so that the edge OX coincides with OY", and the edge OY with OX". If we combine the corresponding edges OX with OX" and OY with OY", then the edges OZ and OZ" will be directed in opposite directions.
If symmetrical figures together make up one geometric body, then they say that this geometric body has a center of symmetry. Thus, if a given body has a center of symmetry, then any point belonging to this body corresponds to a symmetrical point that also belongs to this body. Of the geometric bodies we have considered, for example, the center of symmetry has: 1) a parallelepiped, 2) a prism, which has a regular polygon with an even number of sides at the base.
A regular tetrahedron has no center of symmetry.
100. Symmetry with respect to the plane. Two spatial figures are called symmetrical with respect to the plane P if each point A in one figure corresponds to another point A ", and the segment AA" is perpendicular to the plane P and is divided in half at the point of intersection with this plane.
Theorem. Any two corresponding segments in two symmetrical figures are equal to each other.
Let two figures be given that are symmetrical with respect to the plane P. Let us select two any points A and B of the first figure, let A "and B" be the points of the second figure corresponding to them (Fig. 116, figures are not shown in the drawing).
Let further C be the point of intersection of the segment AA "with the plane P, D - the point of intersection of the segment BB" with the same plane. Connecting points C and D with a straight line segment, we get two quadrilaterals ABDC and A "B" DC. Since AC \u003d A "C, BD \u003d B" D and
/
ACD= /
ACD, /
BDC= /
In "DC, as right angles, then these quadrangles are equal (which is easily verified by superposition). Therefore, AB \u003d AB" It follows directly from this theorem that the corresponding plane and dihedral angles of two figures symmetric with respect to the plane are equal between Nevertheless, it is impossible to combine these two figures with one another so that their corresponding parts are combined, since the order of the parts in one figure is reverse to that in the other (this will be proved below, § 102). two figures that are symmetrical with respect to a plane are: any object and its reflection in a plane mirror; any figure that is symmetrical with its mirror reflection relative to the plane of the mirror.
If any geometric body can be divided into two parts symmetrical with respect to some plane, then this plane is called the plane of symmetry of this body.
Geometric bodies with a plane of symmetry are extremely common in nature and in everyday life. The human and animal body has a plane of symmetry dividing it into right and left parts.
In this example, it is especially clear that symmetrical figures cannot be combined. So, the hands of the right and left hands are symmetrical, but they cannot be combined, which can be seen at least from the fact that the same glove cannot fit both the right and left hands. A large number of household items have a plane of symmetry: a chair, a dining table, a bookcase, a sofa, etc. Some, such as a dining table, even have not one, but two planes of symmetry (Fig. 117).
Usually, when considering an object that has a plane of symmetry, we strive to take such a position in relation to it that the plane of symmetry of our body, or at least our head, coincides with the plane of symmetry of the object itself. In this case. the symmetrical shape of the object becomes especially noticeable.
101. Symmetry about the axis. Axis of symmetry of the second order. Two figures are called symmetrical about the l-axis (the axis is a straight line) if each point A of the first figure corresponds to a point A "of the second figure, so that the segment AA" is perpendicular to the l-axis, intersects with it and is divided in half at the point of intersection. The l-axis itself is called the second-order symmetry axis.
From this definition, it directly follows that if two geometric bodies symmetrical about an axis are intersected by a plane perpendicular to this axis, then two flat figures will be obtained in the section, symmetrical with respect to the point of intersection of the plane with the axis of symmetry of the bodies.
From this, it is further easy to deduce that two bodies symmetrical about an axis can be combined with one another by rotating one of them by 180 ° around the axis of symmetry. Indeed, imagine all possible planes perpendicular to the axis of symmetry.
Each such plane intersecting both bodies contains figures that are symmetrical with respect to the meeting point of the plane with the axis of symmetry of the bodies. If we make the cutting plane slide by itself, rotating it around the axis of symmetry of the body by 180°, then the first figure coincides with the second one.
This is true for any cutting plane. The rotation of all sections of the body by 180° is equivalent to the rotation of the entire body by 180° around the axis of symmetry. This is where the validity of our assertion follows.
If, after the rotation of a spatial figure around a certain straight line by 180 °, it coincides with itself, then they say that the figure has this straight line as its second-order symmetry axis.
The name "axis of symmetry of the second order" is explained by the fact that during a complete rotation around this axis, the body will twice take a position that coincides with the original one (counting the original one) during rotation. Examples of geometric bodies with an axis of symmetry of the second order are:
1) a regular pyramid with an even number of side faces; its axis of symmetry is its height;
2) rectangular parallelepiped; it has three axes of symmetry: straight lines connecting the centers of its opposite faces;
3) a regular prism with an even number of side faces. The axis of its symmetry is each straight line connecting the centers of any pair of its opposite faces (lateral faces and two bases of the prism). If the number of side faces of the prism is 2 k, then the number of such axes of symmetry will be k+ 1. In addition, each straight line connecting the midpoints of its opposite side edges serves as an axis of symmetry for such a prism. A prism has such axes of symmetry.
So the correct 2 k-faceted prism has 2 k+1 axes, symmetry.
102. Dependence between different types of symmetry in space. Between different types of symmetry in space - axial, planar and central - there is a relationship, expressed by the following theorem.
Theorem. If the figure F is symmetrical with the figure F "with respect to the plane P and at the same time symmetrical with the figure F" with respect to the point O lying in the plane P, then the figures F "and F" are symmetrical with respect to the axis passing through the point O and perpendicular to the plane R.
Let's take some point A of figure F (Fig. 118). It corresponds to the point A "of the figure F" and the point A "of the figure F" (the figures themselves F, F" and F" are not shown in the drawing).
Let B be the point of intersection of the segment AA "with the plane P. Let's draw a plane through points A, A" and O. This plane will be perpendicular to the plane P, since it passes through the line AA "perpendicular to this plane. In the plane AA" O we draw straight line OH perpendicular to OB. This line OH will also be perpendicular to the plane P. Further, let C be the intersection point of the lines A"A" and OH.
In the triangle AA "A" "segment BO connects the midpoints of the sides AA" and AA", therefore, BO || A"A", but BO_|_OH, which means A"A"_|_OH. Further, since O is the middle side AA", and CO || AA", then A"C \u003d A"C. From this we conclude that points A" and A" are symmetrical about the axis OH. The same is true for all other points of the figure. Hence, our theorem is proved. It follows directly from this theorem, that two figures symmetrical about a plane cannot be combined so that their respective parts are combined. Indeed, the figure F "is combined with F" by rotating around the OH axis by 180 °. But the figures F "and F cannot be combined as symmetrical with respect to the point, therefore, the figures F and F" cannot be combined either.
103. Axes of symmetry of higher orders. A figure having an axis of symmetry is aligned with itself after being rotated around the axis of symmetry by an angle of 180°. But there are cases when the figure comes to coincide with the initial position after turning around some axis by an angle less than 180°. Thus, if the body makes a complete revolution around this axis, then in the process of rotation it will be combined several times with its original position. Such an axis of rotation is called a higher-order symmetry axis, and the number of body positions that coincide with the original is called the order of the symmetry axis. This axis may not coincide with the axis of symmetry of the second order. So, a regular triangular pyramid does not have a second-order symmetry axis, but its height serves as a third-order symmetry axis for it. Indeed, after turning this pyramid around the height at an angle of 120 °, it is combined with itself (Fig. 119).
When the pyramid rotates around the height, it can occupy three positions, coinciding with the original one, counting the original one as well. It is easy to see that any even-order symmetry axis is at the same time a second-order symmetry axis.
Examples of axes of symmetry of higher orders:
1) Correct n-coal pyramid has an axis of symmetry n-th order. This axis is the height of the pyramid.
2) Correct n-coal prism has an axis of symmetry n-th order. This axis is a straight line connecting the centers of the bases of the prism.
104. The symmetry of the cube. As for any parallelepiped, the point of intersection of the cube's diagonals is the center of its symmetry.
The cube has nine planes of symmetry: six diagonal planes and three planes passing through the midpoints of each four of its parallel edges.
The cube has nine axes of symmetry of the second order: six straight lines connecting the midpoints of its opposite edges, and three straight lines connecting the centers of opposite faces (Fig. 120).
These last lines are axes of symmetry of the fourth order. In addition, the cube has four axes of symmetry of the third order, which are its diagonals. Indeed, the diagonal of the cube AG (Fig. 120) is obviously equally inclined to the edges AB, AD and AE, and these edges are equally inclined to each other. If we connect points B, D and E, we get a regular triangular pyramid ADBE, for which the diagonal of the cube AG serves as a height. When this pyramid aligns with itself as it rotates around the height, the entire cube will align with its original position. It is easy to see that the cube has no other axes of symmetry. Let's see how many different ways a cube can fit into itself. Rotation around the ordinary axis of symmetry gives one position of the cube, different from the original one, in which the cube as a whole is aligned with itself.
Rotation about a 3rd-order axis gives two such positions, and rotation about a 4th-order axis gives three such positions. Since the cube has six axes of the second order (these are ordinary axes of symmetry), four axes of the third order and three axes of the fourth order, there are 6 1 + 4 2 + 3 3 = 23 positions of the cube, different from the original, at which it is combined with yourself.
It is easy to verify directly that all these positions are different from each other, as well as from the initial position of the cube. Together with the original position, they make up 24 ways to combine the cube with itself.
Definition of symmetry;
Definition of symmetry;
Central symmetry;
Axial symmetry;
Symmetry about the plane;
rotational symmetry;
Mirror symmetry;
Symmetry of similarity;
Symmetry of plants;
Animal symmetry;
Symmetry in architecture;
Is man a symmetrical being?
Symmetry of words and numbers;
SYMMETRY
SYMMETRY- proportionality, the sameness in the arrangement of parts of something on opposite sides of a point, line or plane.
(Explanatory dictionary of Ozhegov)
So, a geometric object is considered symmetrical if you can do something with it, after which it will remain unchanged.
O O O called the center of symmetry of the figure.
The figure is called symmetrical with respect to the point O, if for each point of the figure the point symmetrical to it with respect to the point O also belongs to this figure. Dot O called the center of symmetry of the figure.
circle and parallelogram circle center ). Schedule odd function
An example of a figure that does not have a center of symmetry is arbitrary triangle.
Examples of figures that have central symmetry are circle and parallelogram. The center of symmetry of the circle is circle center, and the center of symmetry of the parallelogram is point of intersection of its diagonals. Any line also has central symmetry ( any point of a line is its center of symmetry). Schedule odd function symmetrical about the origin.
a a a called the axis of symmetry of the figure.
The figure is said to be symmetrical with respect to a straight line. a, if for each point of the figure the point symmetrical to it with respect to the straight line a also belongs to this figure. Straight a called the axis of symmetry of the figure.
At an unfolded corner one axis of symmetry angle bisector one axis of symmetry three axes of symmetry on two axes of symmetry, and the square four axes of symmetry relative to the y-axis.
There are figures that do not have any axis of symmetry. These figures include parallelogram, other than a rectangle, scalene triangle.
At an unfolded corner one axis of symmetry- the line on which it is located angle bisector. An isosceles triangle also has one axis of symmetry, and an equilateral triangle three axes of symmetry. A rectangle and a rhombus that are not squares have on two axes of symmetry, and the square four axes of symmetry. A circle has an infinite number of them. The graph of an even function is symmetrical when plotted relative to the y-axis.
points BUT and A1 a a AA1 and perpendicular a counts symmetrical to itself
points BUT and A1 are called symmetric with respect to the plane a(plane of symmetry), if the plane a passes through the middle of the segment AA1 and perpendicular to this segment. Each point of the plane a counts symmetrical to itself. Two figures are said to be symmetric with respect to a plane (or mirror-symmetric with respect to) if they consist of pairwise symmetrical points. This means that for each point of one figure, a (relatively) symmetrical point to it lies in another figure.
The body (or figure) has rotational symmetry, if when turning at an angle 360º/n, where n is an integer fully compatible
The body (or figure) has rotational symmetry, if when turning at an angle 360º/n, where n is an integer, about some straight line AB (axis of symmetry) it fully compatible with its original position.
Radial symmetry- a form of symmetry that is preserved when an object rotates around a certain point or line. Often this point coincides with the center of gravity of the object, that is, the point at which intersects an infinite number of axes of symmetry. Such objects can be circle, ball, cylinder or cone.
Mirror symmetry connects any
Mirror symmetry connects any an 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). Symmetrically mirrored figures, for all their similarities, differ significantly from each other. Two mirror-symmetric flat figures can always be superimposed on each other. However, for this it is necessary to remove one of them (or both) from their common plane.
Similarity symmetry nesting dolls.
Similarity symmetry are peculiar analogues of the previous symmetries, with the only difference that they are associated with simultaneous decrease or increase in similar parts of the figure and the distances between them. The simplest example of such a symmetry is nesting dolls.
Sometimes figures can have different types of symmetry. For example, some letters have rotational and mirror symmetry: F, H, M, O, BUT.
There are many other kinds of symmetries that are abstract in nature. For example:
Permutation symmetry, which consists in the fact that if identical particles are interchanged, then no changes occur;
Gauge symmetries connected with zoom. In inanimate nature, symmetry primarily arises in such a natural phenomenon as crystals that make up almost all solids. It is she who determines their properties. The most obvious example of the beauty and perfection of crystals is the well-known snowflake.
We meet symmetry everywhere: in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. Symmetry principles play an important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music. The laws of nature also obey the principles of symmetry.
axis of symmetry.
Many flowers have an interesting property: they can be rotated so that each petal takes the position of its neighbor, while the flower is combined with itself. This flower has axis of symmetry.
Screw symmetry observed in the arrangement of leaves on the stems of most plants. Being located by a screw along the stem, the leaves seem to spread out in all directions and do not obscure each other from the light, which is essential for plant life.
Bilateral symmetry plant organs also possess, for example, the stems of many cacti. Often found in botany radially symmetrically built flowers.
dividing line.
Symmetry in animals is understood as correspondence in size, shape and shape, as well as the relative location of body parts located on opposite sides. dividing line.
The main types of symmetry are radial(radiation) - it is possessed by echinoderms, coelenterates, jellyfish, etc .; or bilateral(two-sided) - we can say that every animal (be it an insect, a fish or a bird) consists from two halves- right and left.
spherical symmetry occurs in radiolarians and sunflowers. Any plane drawn through the center divides the animal into equal halves.
The symmetry of a structure is associated with the organization of its functions. The projection of the plane of symmetry - the axis of the building - usually determines the location of the main entrance and the beginning of the main traffic flows.
Every detail in a symmetrical system exists as a doppelgänger of his obligatory pair located on the other side of the axis, and due to this it can be considered only as part of the whole.
Most common in architecture mirror symmetry. The buildings of Ancient Egypt and the temples of ancient Greece, amphitheatres, baths, basilicas and triumphal arches of the Romans, palaces and churches of the Renaissance, as well as numerous buildings of modern architecture are subordinate to it.
accents
To better reflect the symmetry on the structures are placed accents- especially significant elements (domes, spiers, tents, main entrances and stairs, balconies and bay windows).
To design the decoration of architecture, an ornament is used - a rhythmically repeating pattern based on the symmetrical composition of its elements and expressed by line, color or relief. Historically, several types of ornaments have developed based on two sources - natural forms and geometric figures.
But an architect is first and foremost an artist. And therefore, even the most "classic" styles often used dissymmetry– a nuanced deviation from pure symmetry, or asymmetry- deliberately asymmetrical construction.
No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same. But the resemblance between our hands, ears, eyes and other parts of the body is the same as between an object and its reflection in a mirror.
right his half rough features characteristic of the male sex. Left half
Numerous measurements of facial parameters in men and women have shown that right his half compared to the left, has more pronounced transverse dimensions, which gives the face a more rough features characteristic of the male sex. Left half face has more pronounced longitudinal dimensions, which gives it smooth lines and femininity. This fact explains the predominant desire of females to pose for artists on the left side of the face, and males on the right.
Palindrome
Palindrome(from Gr. Palindromos - running back) - this is some object in which the symmetry of the components is specified from the beginning to the end and from the end to the beginning. For example, a phrase or text.
The straight text of a palindrome, which is read in accordance with the normal reading direction in a given script (usually from left to right), is called forward, reverse - a shell walker or reverse(from right to left). Some numbers also have symmetry.
So, with regard to geometry: there are three main types of symmetry.
First of all, central symmetry (or symmetry about a point) - this is a transformation of the plane (or space), in which the only point (point O - the center of symmetry) remains in place, while the rest of the points change their position: instead of point A, we get point A1 such that point O is the middle of segment AA1. To construct a figure Ф1, symmetrical to the figure Ф with respect to the point O, it is necessary to draw a ray through each point of the figure Ф passing through the point O (the center of symmetry), and on this ray to set aside a point symmetrical to the one chosen with respect to the point O. The set of points constructed in this way will give a figure F1.
Of great interest are figures that have a center of symmetry: with symmetry about the point O, any point of the figure F is again transformed into some point of the figure F. There are many such figures in geometry. For example: a segment (the middle of the segment is the center of symmetry), a straight line (any of its points is the center of its symmetry), a circle (the center of the circle is the center of symmetry), a rectangle (the point of intersection of its diagonals is the center of symmetry). There are many centrally symmetric objects in animate and inanimate nature (student communication). Often people themselves create objects that have a center of symmetryrii (examples from needlework, examples from mechanical engineering, examples from architecture and many other examples).
Secondly, axial symmetry (or symmetry about a line) - this is a transformation of the plane (or space), in which only the points of the line p remain in place (this line is the axis of symmetry), while the rest of the points change their position: instead of the point B, we get such a point B1 that the line p is the perpendicular bisector to the segment BB1 . To construct a figure Φ1 symmetrical to the figure Φ with respect to the line p, it is necessary for each point of the figure Φ to construct a point symmetric to it with respect to the line p. The set of all these constructed points gives the required figure Ф1. There are many geometric shapes that have an axis of symmetry.
A rectangle has two, a square has four, a circle has any straight line passing through its center. If you look closely at the letters of the alphabet, then among them you can find those that have a horizontal or vertical, and sometimes both axes of symmetry. Objects with axes of symmetry are quite common in animate and inanimate nature (student reports). In his activity, a person creates many objects (for example, ornaments) that have several axes of symmetry.
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Thirdly, planar (mirror) symmetry (or symmetry about a plane)
- this is a transformation of space, in which only points of one plane retain their location (α-plane of symmetry), the remaining points of space change their position: instead of point C, such a point C1 is obtained that the plane α passes through the middle of the segment CC1, perpendicular to it.
To construct a figure Ф1, symmetrical to the figure Ф with respect to the plane α, it is necessary for each point of the figure Ф to build points symmetrical with respect to α, they form the figure Ф1 in their set.
Most often, in the world of things and objects around us, we encounter three-dimensional bodies. And some of these bodies have planes of symmetry, sometimes even several. And the man himself in his activities (construction, needlework, modeling, ...) creates objects with planes of symmetry.
It is worth noting that along with the three listed types of symmetry, there are (in architecture)portable and swivel, which in geometry are compositions of several movements.
Symmetry is associated with harmony and order. And not in vain. Because the question of what symmetry is, there is an answer in the form of a literal translation from ancient Greek. And it turns out that it means proportionality and immutability. And what could be more orderly than a strict definition of location? And what can be called more harmonious than something that strictly corresponds to the size?
What does symmetry mean in different sciences?
Biology. In it, an important component of symmetry is that animals and plants have regularly arranged parts. Moreover, in this science there is no strict symmetry. There is always some asymmetry. It admits that the parts of the whole do not coincide with absolute precision.
Chemistry. Molecules of a substance have a certain regularity in their arrangement. It is their symmetry that explains many properties of materials in crystallography and other branches of chemistry.Physics. The system of bodies and changes in it are described using equations. They contain symmetrical components, which simplifies the whole solution. This is done by searching for conserved quantities.
Mathematics. It is in it that the main explanation is given of what symmetry is. Moreover, it is given more importance in geometry. Here, symmetry is the ability to display in figures and bodies. In a narrow sense, it comes down to just a mirror image.
How do different dictionaries define symmetry?
In whichever of them we look, the word "proportionality" will be found everywhere. In Dahl, one can also see such an interpretation as uniformity and uniformity. In other words, symmetrical means the same. It also says that it is boring, what looks more interesting is what it is not in.
When asked what symmetry is, Ozhegov's dictionary already speaks of the sameness in the position of parts relative to a point, line or plane.
Ushakov's dictionary also mentions proportionality, as well as the full correspondence of the two parts of the whole to each other.
When do people talk about asymmetry?
The prefix "a" negates the meaning of the main noun. Therefore, asymmetry means that the arrangement of elements does not lend itself to a certain pattern. There is no immutability in it.
This term is used in situations where the two halves of the item are not perfectly matched. Most of the time they don't look alike.
In wildlife, asymmetry plays an important role. And it can be both useful and harmful. For example, the heart is placed in the left half of the chest. Due to this, the left lung is significantly smaller. But it is necessary.
About central and axial symmetry
In mathematics, there are such types of it:
- central, that is, performed with respect to one point;
- axial, which is observed near a straight line;
- specular, it is based on reflections;
- transfer symmetry.
What is the axis and center of symmetry? This is a point or line, relative to which any point of the body can find another. Moreover, such that the distance from the original to the resulting one is halved by the axis or center of symmetry. During the movement of these points, they describe the same trajectories.
It is easiest to understand what symmetry about an axis is with an example. The notebook paper should be folded in half. The fold line will be the axis of symmetry. If we draw a perpendicular line to it, then all points on it will have points lying at the same distance on the other side of the axis.
In situations where you need to find the center of symmetry, you need to do the following. If there are two figures, then find the same points for them and connect them with a segment. Then split in half. When the figure is one, then knowledge of its properties can help. Often this center coincides with the point of intersection of the diagonals or heights.
What shapes are symmetrical?
Geometric figures can have axial or central symmetry. But this is not a prerequisite, there are many objects that do not have it at all. For example, a parallelogram has a central but no axial. And non-isosceles trapezoids and triangles have no symmetry at all.
If central symmetry is considered, there are quite a lot of figures possessing it. This is a segment and a circle, a parallelogram and all regular polygons with a number of sides that is divisible by two.
The center of symmetry of a segment (also a circle) is its center, while for a parallelogram it coincides with the intersection of the diagonals. While for regular polygons, this point also coincides with the center of the figure.
If a straight line can be drawn in a figure, along which it can be folded, and the two halves coincide, then it (the straight line) will be the axis of symmetry. What is interesting is how many axes of symmetry different figures have.
For example, an acute or obtuse angle has only one axis, which is its bisector.
If you need to find the axis in an isosceles triangle, then you need to draw the height to its base. The line will be the axis of symmetry. And just one. And in an equilateral one there will be three of them at once. In addition, the triangle also has central symmetry with respect to the point of intersection of the heights.
A circle can have an infinite number of axes of symmetry. Any straight line that passes through its center can fulfill this role.
A rectangle and a rhombus have two axes of symmetry. In the first one, they pass through the midpoints of the sides, and in the second, they coincide with the diagonals.
The square combines the previous two figures and has 4 axes of symmetry at once. They are the same as those of a rhombus and a rectangle.
