How to find the axis of symmetry of a segment. Axes of symmetry

Construct a segment A1B1 symmetrical to segment AB with respect to point O. Point O is the center of symmetry. A1. V. O. A. Note: with symmetry about the center, the order of points has changed (top-bottom, right-left). For example, point A is displayed from bottom to top; it was to the right of point B, and its image point A1 turned out to be to the left of point B1.

slide 16 from the presentation "Symmetry of figures". The size of the archive with the presentation is 680 KB.

Geometry Grade 9

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The purpose of the lesson:

formation of the concept of "symmetrical points";
teach children to build points that are symmetrical to data;
learn to build segments symmetrical to data;
consolidation of the past (formation of computational skills, dividing a multi-digit number into a single-digit one).

On the stand "to the lesson" cards:

1. Organizational moment

Greetings.

The teacher draws attention to the stand:

Children, we begin the lesson by planning our work.

Today at the lesson of mathematics we will take a trip to 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 will 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 put a bench, and to the left of the bench and to the right of it at the same distance he put exactly the same armfuls of hay.

Figure 1 on the board:

The donkey walked from one armful of hay to another, but did not decide which armful to start with. And, in the end, he died of hunger.

Why didn't the donkey decide which handful 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 some 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 to make sure that they are really symmetrical? (fold the sheet, the points 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 it?

(Measure distance from straight line to points)

We return to our pieces of colored paper.

Measure the distance from the fold line (axis of symmetry), first to one and then to another point (but first connect them with a segment).

What can you say about these distances?

(The same)

Find the midpoint of your segment.

Where is she?

(It is the point of intersection of the segment AB with the axis of symmetry)

4. Pay attention to the corners, formed as a result of the intersection of the segment AB with the axis of symmetry. (We find out with the help of a square, each child works at his workplace, one studies on the board).

Conclusion of children: 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 line or axis of symmetry, then the segment connecting these points is at a right angle, or perpendicular to this line. (The word "perpendicular" is written separately on the stand). The word "perpendicular" is pronounced aloud in unison.

5. Let's pay attention to how this rule is written in our textbook.

Textbook work.

Find symmetrical points about a straight line. Will points A and B be symmetrical about this line?

6. Working on new material.

Let's learn how to build points that are symmetrical to data about a straight line.

The teacher teaches to reason.

To construct a point symmetrical to point A, you need to move this point from the line by the same distance to the right.

7. We will learn to build segments that are symmetrical to data, relative to a straight line. Textbook work.

Students discuss at the blackboard.

8. Oral account.

On this we will finish our stay in the "Geometry" Kingdom and conduct a small mathematical warm-up, having visited the "Arithmetic" kingdom.

While everyone is working orally, two students work on individual boards.

A) Perform a division with a check:

B) After inserting the necessary numbers, solve the example and check:

Verbal counting.

The life expectancy of a birch is 250 years, and an oak is 4 times longer. How many years 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 called guests to his place: 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

(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 right number:

3 9 81
2 16

5 10 20
6 24

9. And now let's rest a little.

Listen to Beethoven's Moonlight Sonata. A moment of classical music. Students put their heads on the desk, close their eyes, listen to music.

10. Journey into the realm of algebra.

Guess the roots of the equation and check:

Students decide on the board and in notebooks. Explain how you figured it out.

11. "Blitz tournament" .

a) Asya bought 5 bagels for a rubles and 2 loaves for b rubles. How much does the whole purchase cost?

We check. We share opinions.

12. Summarizing.

So, we have completed our journey into the realm of mathematics.

What was the most important thing for you in the lesson?

Who liked our lesson?

I enjoyed working with you

Thank you for the lesson.

Human life is filled with symmetry. It is convenient, beautiful, no need to invent new standards. But what is she really and is she as beautiful in nature as is commonly believed?

Symmetry

Since ancient times, people have sought to streamline the world around them. Therefore, something is considered beautiful, and something not so. From an aesthetic point of view, golden and silver sections are considered attractive, as well as, of course, symmetry. This term has Greek origin and literally means "proportion". Of course we are talking not only about the coincidence on this basis, but also on some others. In a general sense, symmetry is such 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 animate 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 value remains generally unchanged. This phenomenon is quite common 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 ornaments on fabric, building borders and many others. man-made objects. It is worth considering this phenomenon in more detail, because it is extremely exciting.

Use of the term in other scientific fields

In the following, symmetry will be considered from the point of view of geometry, but it is worth mentioning that given word used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon is studied from various angles and in different conditions. The classification, for example, depends on which science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged everywhere.

Classification

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

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

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

In biology, all species are called somewhat differently, although in fact they can be the same. The division into certain groups occurs on the basis of the presence or absence, as well as the number of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

Some features are distinguished in the phenomenon, one of which is necessarily present. The so-called 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 called the point inside the figure or crystal, at which the lines converge, connecting in pairs all sides parallel to each other. Of course, it doesn't always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since there is none. According to the definition, it is obvious that the center of symmetry is that through which the figure can be reflected to itself. An example is, for example, a circle and a point in its middle. This element is usually referred to as C.

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

But perhaps the most common is what is called "axes of symmetry." This frequent phenomenon can be seen both in geometry and in nature. And it deserves separate consideration.

axes

Often the element with respect to which the figure can be called symmetrical,

is a straight line or a segment. 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: divide sides or be parallel to them, as well as cross corners or not. Axes of symmetry are usually denoted as L.

Examples are isosceles and In the first case it will be vertical axis symmetry, on both sides of which there are equal faces, and in the second line will intersect each corner and coincide with all bisectors, medians and heights. Ordinary triangles do not have it.

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

It is conditionally possible to divide the entire set of objects of study of mathematicians into figures that have an axis of symmetry, and those that do not. All circles, ovals, as well as some special cases automatically fall into the first category, while the rest fall into the second group.

As in the case when it was said about the axis of symmetry of the triangle, this element for the quadrilateral does not always exist. For a square, rectangle, rhombus or parallelogram, it is, but for irregular shape, respectively, no. For a circle, the axis of symmetry is the set of straight lines that pass through its center.

Moreover, it is interesting to consider three-dimensional figures from this point of view. At least one axis of symmetry, in addition to all regular polygons and the ball, will have some cones, as well as pyramids, parallelograms and some others. Each case must be considered separately.

Examples in nature

In life it is called bilateral, it occurs most
often. Any person and very many animals are an example of this. The axial one is called radial and is much less common, as a rule, in flora. And yet they are. For example, it is worth considering how many axes of symmetry a star has, and does it have them at all? Of course, we are talking about marine life, and not about the subject of study of astronomers. And the correct answer would be this: it depends on the number of rays of the star, for example, five, if it is five-pointed.

In addition, radial symmetry is observed in many flowers: chamomile, cornflowers, sunflowers, etc. Examples great amount 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. AT this case a 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 be a beautiful device, for example, in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous one is slightly inclined, and although it is not the only one, 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 also not completely symmetrical. There have even been studies, according to the results of which the "correct" faces were regarded as inanimate or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore extremely interesting.