How many different axes of symmetry a triangle can have depends on its geometric shape. If this is an equilateral triangle, then it will have as many as three axes of symmetry.

And if it is an isosceles triangle, it will have only one axis of symmetry.

My sister's son is studying this topic in geometry lessons at school. The axis of symmetry is a straight line, when rotated around which by a specific angle, a symmetrical figure will take the same position in space that it occupied before the rotation, and some of its parts will be replaced by the same others. In an isosceles triangle there are three, in a right triangle there is one, in the others there are none, since their sides are not equal to each other.

It depends on what kind of triangle it is. An equilateral triangle has three axes of symmetry that pass through its three vertices. An isosceles triangle, accordingly, has one axis of symmetry. The remaining triangles do not have axes of symmetry.



The simplest thing you can remember is that an equilateral triangle has three equal sides and has three axes of symmetry

This makes it easier to remember the following

There are no equal sides, that is, all sides are different, which means there are no axes of symmetry

And in an isosceles triangle there is only one axis



You cannot simply answer how many axes of symmetry a triangle has without understanding which particular triangle we are talking about.

An equilateral triangle has three axes of symmetry, respectively.

An isosceles triangle has only one axis of symmetry.

Any other triangles with sides of different lengths do not have any axis of symmetry at all.

A triangle in which all sides are different in size does not have axes of symmetry.

A right triangle can have one axis of symmetry if its legs are equal.

In a triangle in which two sides are equal (isosceles), one axis can be drawn, and in which all three sides are equal (equilateral) - three.

Before answering the question of how many axes of symmetry a triangle has, you first need to remember what an axis of symmetry is.

So, to put it simply, in geometry, the axis of symmetry is a line along which if you bend a figure, you get identical halves.

but it is worth remembering that triangles are also different.

So, isosceles triangle (triangle with two equal sides) has one axis of symmetry.

Equilateral a triangle accordingly has 3 axes of symmetry, since all sides of this triangle are equal.

And here versatile A triangle has no axes of symmetry at all. No matter how you fold it and no matter where you draw straight lines, but since the sides are different, you won’t get two identical halves.



As far as I remember geometry, an equilateral triangle has three axes of symmetry passing through its vertices, these are its bisectors. U right triangle, like scalene, obtuse and acute triangles, there are no axes of symmetry at all, but an isosceles triangle has one.

And it’s easy to check - just imagine a line along which it can be cut in half so as to get two identical triangles.

Since triangles are different, they also have axes of symmetry in different quantities. For example, a triangle with different sides has no axes of symmetry at all. And the equilateral has three of them. There is another type of triangle that has one axis of symmetry. It has two equal sides and one right angle.

An arbitrary triangle has no axes of symmetry. An isosceles triangle has one axis of symmetry - the median to the single side. An equilateral triangle has three axes of symmetry - these are its three medians.

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 on 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.

– Let’s consider volumetric 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, in 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.

Today we will talk about a phenomenon that each of us constantly encounters in life: symmetry. What is symmetry?

We all roughly understand the meaning of this term. The dictionary says: symmetry is proportionality and complete correspondence of the arrangement of parts of something relative to a straight line or point. There are two types of symmetry: axial and radial. Let's look at the axial one first. This is, let’s say, “mirror” symmetry, when one half of an object is completely identical to the second, but repeats it as a reflection. Look at the halves of the sheet. They are mirror symmetrical. The halves of the human body are also symmetrical (front view) - identical arms and legs, identical eyes. But let’s not be mistaken; in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet copy each other far from perfectly, the same applies to the human body (take a closer look for yourself); The same is true for other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer only in one position. It’s worth, say, turning a sheet of paper, or raising one hand, and what happens? – you see for yourself.

People achieve true symmetry in the works of their labor (things) - clothes, cars... In nature, it is characteristic of inorganic formations, for example, crystals.

But let's move on to practice. You shouldn’t start with complex objects like people and animals; let’s try to finish drawing the mirror half of the sheet as the first exercise in a new field.

Drawing a symmetrical object - lesson 1

We make sure that it turns out as similar as possible. To do this, we will literally build our soul mate. Don’t think that it’s so easy, especially the first time, to draw a mirror-corresponding line with one stroke!

Let's mark several reference points for the future symmetrical line. We proceed like this: with a pencil, without pressing, we draw several perpendiculars to the axis of symmetry - the midrib of the leaf. Four or five is enough for now. And on these perpendiculars we measure to the right the same distance as on the left half to the line of the edge of the leaf. I advise you to use a ruler, don’t rely too much on your eye. As a rule, we tend to reduce the drawing - this has been observed from experience. We do not recommend measuring distances with your fingers: the error is too large.

Let's connect the resulting points with a pencil line:

Now let’s look meticulously at whether the halves are really the same. If everything is correct, we will circle it with a felt-tip pen and clarify our line:

The poplar leaf has been completed, now you can take a swing at the oak leaf.

Let's draw a symmetrical figure - lesson 2

In this case, the difficulty lies in the fact that the veins are marked and they are not perpendicular to the axis of symmetry and not only the dimensions but also the angle of inclination will have to be strictly observed. Well, let’s train our eye:

So a symmetrical oak leaf has been drawn, or rather, we built it according to all the rules:

How to draw a symmetrical object - lesson 3

And let’s consolidate the theme - we’ll finish drawing a symmetrical lilac leaf.

He has too interesting shape- heart-shaped and with ears at the base, you’ll have to puff:

This is what they drew:

Take a look at the resulting work from a distance and evaluate how accurately we were able to convey the required similarity. Here's a tip: look at your image in the mirror and it will tell you if there are any mistakes. Another way: bend the image exactly along the axis (we have already learned how to bend it correctly) and cut out the leaf along the original line. Look at the figure itself and at the cut paper.

An axis of symmetry is a straight line, when rotated around it through a certain angle, the figure aligns with itself.

The smallest angle of rotation that brings the figure into self-alignment is called elementary axis rotation angle. The elementary rotation angle of the axis  is an integer times 360 :

where n is an integer.

The number n, showing how many times the elementary angle of rotation of the axis is contained in 360 0, is called axis order.

Geometric figures can contain axes of any order, starting from an axis of the first order and ending with an axis of infinite order.

The elementary angle of rotation of the first order axis (n = 1) is equal to 360 0. Since each figure, being rotated around any direction by 360 0, is combined with itself, then each figure has an infinite number of first-order axes. Such axes are not characteristic, so they are usually not mentioned.

An axis of infinite order corresponds to an infinitely small elementary angle of rotation. This axis is present in all rotation figures as the axis of rotation.

Examples of axes of the third, fourth, fifth, sixth, etc. orders are perpendiculars to the drawing plane, passing through the centers of regular polygons, triangles, squares, pentagons, etc.

Thus, in geometry there is an infinite number of axes of different orders.

In crystalline polyhedra, not any symmetry axes are possible, but only axes of the first, second, third, fourth and sixth order.

Symmetry axes of the fifth and higher than the sixth order are impossible in crystals. This position is one of the basic laws of crystallography and is called the law of symmetry of crystals.

Like other geometric laws of crystallography, the law of crystal symmetry is explained by the lattice structure of the crystalline substance. Indeed, since the symmetry of a crystal is a manifestation of the symmetry of its internal structure, then only such symmetry elements are possible in crystals that do not contradict the properties of the spatial lattice.

Let us prove that the fifth order axis does not satisfy the laws of the spatial lattice and thereby prove its impossibility in crystalline polyhedra.

Let us assume that a fifth order axis in the spatial lattice is possible. Let this axis be perpendicular to the drawing plane, intersecting it at point O (Fig. 2.9). In a particular case, point O may coincide with one of the lattice nodes.

Rice. 2.9. A symmetry axis of the fifth order is impossible in spatial lattices

Let's take the lattice node A 1 closest to the axis, lying in the plane of the drawing. Since everything is repeated five times around the fifth-order axis, there should be only five nodes closest to it in the drawing plane: A 1, A 2, A 3, A 4, A 5. Located at equal distances from point O at the vertices of a regular pentagon, they are aligned with each other when rotated around O by 360/5 = 72°.

These five nodes, lying in the same plane, form a flat mesh of the spatial lattice and therefore all the basic properties of the lattice are applicable to them. If nodes A 1 and A 2 belong to a row of a flat grid with a gap A 1 A 2, then through any lattice node you can draw a row parallel to row A 1 A 2. Let's draw such a row through node A 3. This row, which also passes through node A 5, must have a gap equal to A 1 A 2, since in a spatial lattice all parallel rows have the same density.

Therefore, at a distance A 3 A x = A 1 A 2 from node A 3 there must be another node A x. However, the additional node A x turns out to lie closer to point O than node A 1, taken by condition to be closest to the fifth-order axis.

Thus, the assumption we made about the possibility of a fifth-order axis in spatial lattices led us to obvious absurdity and is therefore erroneous.

Since the existence of a fifth-order axis is incompatible with the basic properties of the spatial lattice, such an axis is impossible in crystals.

In a similar way, the impossibility of the existence of symmetry axes higher than the sixth order in crystals is proven and, conversely, the possibility of axes of the second, third, fourth and sixth order in crystals, the presence of which does not contradict the properties of spatial lattices.

To designate axes of symmetry, the letter L is used, and the order of the axis is indicated by a small number located to the right of the letter (for example, L 4 is a fourth-order axis).

In crystalline polyhedra, symmetry axes can pass through the centers of opposite faces perpendicular to them, through the midpoints of opposite edges perpendicular to them (only L 2) and through the vertices of the polyhedron. In the latter case, symmetrical faces and edges are equally inclined to a given axis.

A crystal can have several symmetry axes of the same order, the number of which is indicated by the coefficient in front of the letter. For example, in a rectangular parallelepiped there is 3L 2, i.e., three axes of symmetry of the second order; in the cube there are 3L 4, 4L 3 and 6L 2, i.e. three axes of symmetry of the fourth order, four axes of the third order and six axes of the second order, etc.