Axes of symmetry of a regular triangle. Does a triangle have a center of symmetry?

There are two types of symmetry: central and axial. With central symmetry, any straight line drawn through the center of the figure divides it into two absolutely identical parts that are completely symmetrical. In simple words, they are mirror images of each other. An infinite number of such lines can be drawn around a circle; in any case, they will divide it into two symmetrical parts.

Axis of symmetry

Most geometric shapes do not have such characteristics. Only the axis of symmetry can be drawn in them, and not for everyone. An axis is also a straight line that divides a figure into symmetrical parts. But for the axis of symmetry there is only a certain location and if it is slightly changed, the symmetry will be broken.

It is logical that each square has an axis of symmetry, because all its sides are equal and each angle is ninety degrees. Triangles are different. Triangles, in which all sides are different, cannot have either an axis or a center of symmetry. But in isosceles triangles you can draw an axis of symmetry. Recall that an isosceles triangle is considered to have two equal sides and, accordingly, two equal angles adjacent to the third side - the base. For an isosceles triangle, the axis will be a straight line passing from the vertex of the triangle to the base. In this case, this line will be both a median and a bisector, since it will divide the angle in half and reach exactly the middle of the third side. If you fold a triangle along this straight line, the resulting figures will completely copy each other. However, in an isosceles triangle there can be only one axis of symmetry. If we draw another straight line through its center, it will not divide it into two symmetrical parts.

Special triangle

The equilateral triangle is unique. This is a special type of triangle, which is also isosceles. True, each side of it can be considered a base, since all its sides are equal, and each angle is sixty degrees. Therefore, an equilateral triangle has three axes of symmetry. These lines converge at one point in the center of the triangle. But even this feature does not transform an equilateral triangle into a figure with central symmetry. Even an equilateral triangle does not have a center of symmetry, since through the indicated point only three straight lines divide the figure into equal parts. If you draw a straight line in a different direction, then the triangle will no longer have symmetry. This means that these figures have only axial symmetry.

If all the angles in a quadrilateral are right angles, then it is called a rectangle.

Figure 125 shows rectangle ABCD.

Sides AB and BC have a common vertex B. They are called neighboring sides of rectangle ABCD. Also adjacent are, for example, sides BC and CD.

The adjacent sides of a rectangle are called length And width.

Sides AB and CD do not have common vertices. They are called opposite sides of rectangle ABCD. Also opposite are sides BC and AD.

The opposite sides of a rectangle are equal.

In Figure 125, AB = CD, BC = AD. If the length of a rectangle is a and its width is b, then its perimeter is calculated using the formula already familiar to you:

P = 2 a + 2 b

A rectangle with all sides equal is called square(Fig. 126).

Let us draw a straight line l passing through the midpoints of two opposite sides of the rectangle (Fig. 127). If a sheet of paper is folded along a straight line l, then the two parts of the rectangle lying on opposite sides of the straight line l will coincide.

The figures shown in Figure 128 have a similar property. Such figures are called symmetrical about a straight line . The straight line l is called axis of symmetry of the figure .

So, a rectangle is a figure that has an axis of symmetry. Also, the axis of symmetry has an isosceles triangle (Fig. 129).

A figure can have more than one axis of symmetry. For example, a rectangle other than a square has two axes of symmetry (Fig. 130), and a square has four axes of symmetry (Fig. 131). An equilateral triangle has three axes of symmetry (Fig. 132).

While studying the world around us, we often encounter symmetry. Examples of symmetry in nature are shown in Figure 133.

Objects that have an axis of symmetry are easy to perceive and pleasing to the eye. It is not without reason that in Ancient Greece the word “symmetry” served as a synonym for the words “harmony” and “beauty”.

The idea of ​​symmetry is widely used in fine arts and architecture (Fig. 134).

Goals:

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

DURING THE CLASSES

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

Exercise 1(3 min).

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

Question: What function does this line serve?

Suggested answer: This line divides the figure in half.

Question: How are all the points of the figure located on the two resulting halves?

Suggested answer: All points of the halves are at an equal distance from the fold line and at the same level.

– This means that the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is an axis of symmetry.

Task 2 (2 minutes).

– Cut out a snowflake, find the axis of symmetry, characterize it.

Task 3 (5 minutes).

– Draw a circle in your notebook.

Question: Determine how the axis of symmetry goes?

Suggested answer: Differently.

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

Suggested answer: A lot of.

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

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

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

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

I distribute halves of plasticine figures to students.

Task 4 (3 min).

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

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

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

Task 5 (group work 5 min).

– Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.

The correctness of the work performed is determined by the students themselves.

Elements of drawings are presented to students

Task 6 (2 minutes).

– Find the symmetrical parts of these drawings.

To consolidate the material covered, I suggest the following tasks, scheduled for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What type of triangles are these?

2. Draw several isosceles triangles in your notebook with a common base of 6 cm.

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

– Our initial ideas about form date back to the very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new Stone Age, the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
– Where does symmetry occur in nature?

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

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

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

Homework:

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

Axial symmetry is symmetry about a straight line.

Let some straight line be given g.

To construct a point symmetrical to some point A relative to a straight line g, necessary:

1) Draw from point A to a straight line g perpendicular to AO.

2) On the continuation of the perpendicular on the other side of the line g set aside segment OA1 equal to segment AO: OA1=AO.

The resulting point A1 is symmetrical to point A relative to the straight line g.

Straight g called the axis of symmetry.

Thus, points A and A1 are symmetrical with respect to line g if this line passes through the middle of segment AA1 and is perpendicular to it.

If a point A lies on a line g, then the point symmetric to it is the point A itself.

Transformation of figure F into figure F1, in which each of its points A goes to point A1, symmetrical with respect to a given line g, is called a symmetry transformation about a line g.

Figures F and F1 are called figures symmetrical about a straight line g.

To construct a triangle symmetrical to a given one with respect to a straight line g, it is enough to construct points symmetrical to the vertices of the triangle and connect them with segments.

For example, triangles ABC and A1B1C1 are symmetrical about a straight line g.

If the symmetry transformation is relative to the straight line g translates a figure into itself, then such a figure is called symmetrical with respect to a straight line g, and the straight line g is called its axis of symmetry.

A symmetrical figure is divided by its axis of symmetry into two equal halves. If you draw a symmetrical figure on paper, cut it out and bend it along the axis of symmetry, then these halves will coincide.

Examples of figures that are symmetrical about a straight line.

1) Rectangle.

A rectangle has 2 axes of symmetry: straight lines passing through the intersection point of the diagonals parallel to the sides.

A rhombus has two axes of symmetry:

the lines on which its diagonals lie.

3) A square, like a rhombus and a rectangle, has four axes of symmetry: straight lines containing its diagonals, and straight lines passing through the intersection point of the diagonals parallel to the sides.

4) Circle.

A circle has an infinite number of axes of symmetry:

any straight line containing the diameter is the axis of symmetry of the circle.

A straight line also has an infinite number of axes of symmetry: any straight line perpendicular to it is an axis of symmetry for a given straight line.

6) Isosceles trapezoid.

An isosceles trapezoid is a figure that is symmetrical about a straight line, perpendicular to the bases and passing through their midpoints.

7) Isosceles triangle.

An isosceles triangle has one axis of symmetry:

a straight line passing through the height (median, bisector) drawn to the base.

8) Equilateral triangle.

An equilateral triangle has three axes of symmetry:

An angle is a figure that is symmetrical with respect to the straight line containing its bisector.

Axial symmetry is movement.

Symmetry

Since ancient times, people have sought to organize the world around them. Therefore, some things are considered beautiful, and some are not so much. From an aesthetic point of view, the golden 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 original data. It 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 occurs quite often and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in patterns on fabric, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely fascinating.

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 what science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged throughout.

Classification

There are several main types of symmetry, of which three are the most common:

In addition, the following types are also distinguished in geometry; they are much less common, but no less interesting:

• sliding;
• rotational;
• point;
• progressive;
• screw;
• fractal;
• etc.

In biology, all species are called slightly differently, although in essence they may be the same. Division into certain groups occurs on the basis of the presence or absence, as well as the quantity of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

The phenomenon has certain features, one of which is necessarily present. The so-called basic 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 the lines connecting in pairs all sides parallel to each other converge. 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. According to the definition, it is obvious that the center of symmetry is that through which a figure can be reflected onto itself. An example would be, for example, a circle and a point in its middle. This element is usually designated as C.

The plane of symmetry, of course, is imaginary, but it is precisely it that divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or divide them. For the same figure, several planes can exist at once. These elements are usually designated as P.

But perhaps the most common is what is called “axis of symmetry”. This is a common phenomenon that can be seen both in geometry and in nature. And it is worthy of separate consideration.

Axles

Often the element in relation to which a figure can be called symmetrical is

a straight line or segment appears. In any case, we are not talking about a point or a plane. Then the axes of symmetry of the figures are considered. There can be a lot of them, and they can be located in any way: dividing the sides or being parallel to them, as well as intersecting corners or not doing so. Axes of symmetry are usually designated as L.

Examples include isosceles and equilateral triangles. 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 altitudes. Ordinary triangles do not have this.

By the way, the totality of all the above 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, we can divide the entire set of objects of study by mathematicians into figures that have an axis of symmetry and those that do not. All regular polygons, 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 we talked about the axis of symmetry of a triangle, this element does not always exist for a quadrilateral. For a square, rectangle, rhombus or parallelogram it is, but for an irregular figure, accordingly, it is not. For a circle, the axes of symmetry are the set of straight lines that pass through its center.

In addition, it is interesting to consider three-dimensional figures from this point of view. In addition to all regular polygons and the 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

Mirror symmetry in life is called bilateral, it is most common
often. Any person and many animals are an example of this. The axial one is called radial and is found much less frequently, as a rule, in the plant world. And yet they exist. For example, it is worth thinking about how many axes of symmetry a star has, and does it have any 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: 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: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.

Arrhythmia

This term, first of all, reminds most of medicine and cardiology, but 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 found as an accident, and sometimes it can become a wonderful technique, for example in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous Leaning Tower of Pisa is slightly tilted, and although it is not the only one, it is the most famous example. It is known that this happened by accident, but this has its own charm.

In addition, it is obvious that the faces and bodies of people and animals are not completely symmetrical either. There have even been studies where “correct” faces were judged as lifeless or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore are extremely interesting.

Geometric symmetry

When applied to a geometric figure, symmetry means that if this figure is transformed - for example, rotated - some of its properties will remain the same.

The possibility of such transformations varies from figure to figure. For example, a circle can be rotated as much as you like around a point located in its center, it will remain a circle, nothing will change for it.

The concept of symmetry can be explained without resorting to rotation. It is enough to draw a straight line through the center of the circle and construct a segment perpendicular to it anywhere in the figure, connecting two points on the circle. The point of intersection with the line will divide this segment into two parts, which will be equal to each other.

In other words, the straight line divided the figure into two equal parts. The points of the parts of the figure located on lines perpendicular to the given one are at an equal distance from it. This straight line will be called the axis of symmetry. Symmetry of this kind - relatively straight - is called axial symmetry.

Number of axes of symmetry

For different figures, the number of axes of symmetry will be different. For example, a circle and a ball have many such axes. An equilateral triangle has an axis of symmetry that is perpendicular to each side; therefore, it has three axes. A square and a rectangle can have four axes of symmetry. Two of them are perpendicular to the sides of the quadrilaterals, and the other two are diagonals. But an isosceles triangle has only one axis of symmetry, located between its equal sides.

Axial symmetry also occurs in nature. It can be observed in two versions.

The first type is radial symmetry, which involves the presence of several axes. It is typical, for example, for starfish. More highly developed organisms are characterized by bilateral, or bilateral symmetry with a single axis dividing the body into two parts.

The human body also has bilateral symmetry, but it cannot be called ideal. The legs, arms, eyes, lungs are located symmetrically, but not the heart, liver or spleen. Deviations from bilateral symmetry are noticeable even externally. For example, it is extremely rare for a person to have identical moles on both cheeks.

People's lives are filled with symmetry. It’s convenient, beautiful, and there’s no need to invent new standards. But what is it really and is it as beautiful in nature as is commonly believed?

Symmetry

Since ancient times, people have sought to organize the world around them. Therefore, some things are considered beautiful, and some are not so much. From an aesthetic point of view, the golden 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 original data. It 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 occurs quite often and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in patterns on fabric, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely fascinating.

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 what science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged throughout.

Classification

There are several main types of symmetry, of which three are the most common:

In addition, the following types are also distinguished in geometry; they are much less common, but no less interesting:

• sliding;
• rotational;
• point;
• progressive;
• screw;
• fractal;
• etc.

In biology, all species are called slightly differently, although in essence they may be the same. Division into certain groups occurs on the basis of the presence or absence, as well as the quantity of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

The phenomenon has certain features, one of which is necessarily present. The so-called basic 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 the lines connecting in pairs all sides parallel to each other converge. 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. According to the definition, it is obvious that the center of symmetry is that through which a figure can be reflected onto itself. An example would be, for example, a circle and a point in its middle. This element is usually designated as C.

The plane of symmetry, of course, is imaginary, but it is precisely it that divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or divide them. For the same figure, several planes can exist at once. These elements are usually designated as P.

But perhaps the most common is what is called “axis of symmetry”. This is a common phenomenon that can be seen both in geometry and in nature. And it is worthy of separate consideration.

Axles

Often the element in relation to which a figure can be called symmetrical is

a straight line or segment appears. 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 in any way: dividing the sides or being parallel to them, as well as intersecting corners or not doing so. Axes of symmetry are usually designated as L.

Examples include 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 altitudes. Ordinary triangles do not have this.

By the way, the totality of all the above 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, we can divide the entire set of objects of study by 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 we talked about the axis of symmetry of a triangle, this element does not always exist for a quadrilateral. 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 three-dimensional figures from this point of view. In addition to all regular polygons and the 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 one is called radial and is found much less frequently, as a rule, in the plant world. And yet they exist. For example, it is worth thinking about how many axes of symmetry a star has, and does it have any 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: 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: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.

Arrhythmia

This term, first of all, reminds most of medicine and cardiology, but 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 found as an accident, and sometimes it can become a wonderful technique, for example in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous one is slightly tilted, and although it is not the only one, it is the most famous example. It is known that this happened by accident, but this has its own charm.

In addition, it is obvious that the faces and bodies of people and animals are not completely symmetrical either. There have even been studies that show that “correct” faces are judged to be lifeless or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore are extremely interesting.