Like geometric shapes. Basic geometric concepts. Geometric solids

Geometric figure defined as any set of points.

If all points of a geometric figure belong to the same plane, it is called flat. For example, a segment, a rectangle are flat figures. There are figures that are not flat. This is, for example, a cube, a ball, a pyramid.

Since the concept of a geometric figure is defined through the concept of a set, we can say that one figure is included in another (or is contained in another), we can consider the union, intersection and difference of figures.

The point is an indefinable concept. The point is usually introduced by drawing it or piercing it with a pen in a piece of paper. A point is considered to have neither length, nor width, nor area.

Line is an undefined concept. They introduce the line by modeling it from a cord or drawing it on a board, on a piece of paper. The main property of a straight line: a straight line is infinite. Curved lines can be closed or open.

Ray is a part of a straight line bounded on one side.

Line segment- the part of a straight line enclosed between two points - the ends of the segment.

broken line- a line of segments connected in series at an angle to each other. The link of a broken line is a segment. The connection points of the links are called the vertices of the polyline.

Corner- This is a geometric figure that consists of a point and two rays emanating from this point. The rays are called the sides of the angle, and their common beginning is its vertex. An angle is denoted in different ways: either its vertex, or its sides, or three points are indicated: the vertex and two points on the sides of the angle.

An angle is called straight if its sides lie on the same straight line. An angle that is half a straight angle is called a right angle. An angle less than a right angle is called an acute angle. An angle greater than a right angle but less than a straight angle is called an obtuse angle.

Two angles are called adjacent if they have one side in common and the other sides of these angles are complementary half-lines.

Triangle is one of the simplest geometric shapes. A triangle is a geometric figure, which consists of three points that do not lie on one straight line, and three pairwise segments connecting them. In any triangle, the following elements are distinguished: sides, angles, heights, bisectors, medians, midlines.

An acute triangle is a triangle in which all angles are acute. Right Angle - A triangle that has a right angle. A triangle that has an obtuse angle is called an obtuse triangle. Triangles are said to be congruent if their corresponding sides and corresponding angles are equal. In this case, the corresponding angles must lie against the corresponding sides. A triangle is called isosceles if its two sides are equal. These equal sides are called the sides, and the third side is called the base of the triangle.

quadrilateral A figure is called a figure that consists of four points and four segments connecting them in series, and no three of these points should lie on one straight line, and the segments connecting them should not intersect. These points are called vertices of the quadrilateral, and the segments connecting them are called sides.

A diagonal is a line segment connecting opposite vertices of a polygon.

Rectangle A quadrilateral is called in which all angles are right.

Square m is a rectangle in which all sides are equal.

polygon is called a simple closed broken line if its adjacent links do not lie on the same straight line. The vertices of the polyline are called the vertices of the polygon, and its links are called its sides. Segments connecting non-neighbors are called diagonals.

circumference called a figure that consists of all points of the plane equidistant from a given point, which is called the center. But since this classical definition is not given in the elementary grades, acquaintance with the circle is carried out by the method of display, connecting it with direct practical activity in drawing a circle with a compass. The distance from the points to its center is called the radius. A line segment connecting two points on a circle is called a chord. The chord passing through the center is called the diameter.

A circle the part of a plane bounded by a circle.

Parallelepiped A prism whose base is a parallelogram.

Cube is a rectangular parallelepiped, all edges of which are equal.

Pyramid- a polyhedron, in which one face (it is called the base) is some kind of polygon, and the remaining faces (they are called lateral) are triangles with a common vertex.

Cylinder- a geometric body formed by segments of all parallel lines enclosed between two parallel planes, intersecting the circle in one of the planes, and perpendicular to the planes of the bases. A cone is a body formed by all segments connecting a given point - its top - with points of a certain circle - the base of the cone.

Ball is the set of points in space located at a distance not greater than some given positive distance from a given point. The given point is the center of the ball, and the given distance is the radius.

In the lesson you will learn what geometric shapes are. We will talk about the figures depicted on the plane, their properties. You will learn about such simple forms of geometric figures as a point and a line. Consider how a line segment and a ray are formed. Get to know the definition and different types of angles. The next figure, the definition and properties of which are discussed in the lesson, is a circle. Next, the definition of triangle and polygon and their variations are discussed.

Rice. 10. Circle and circumference

Think about which points belong to the circle and which circles (see Fig. 11).

Rice. 11. Mutual arrangement of points and a circle, points and a circle

The correct answer is: points, belong to a circle, and only points and belong to a circle.

A point is the center of a circle or circle. Segments are the radii of a circle or circle, that is, segments that connect the center and any point lying on the circle. A segment is the diameter of a circle or circle, that is, it is a segment connecting two points lying on a circle and passing through the center. The radius is half the diameter (see Fig. 12).

Rice. 12. Radius and diameter

Let's now remember what shape is called a triangle. A triangle is a geometric figure consisting of three points that do not lie on the same straight line, and three line segments connecting these points in pairs. The triangle has three corners.

Consider a triangle (see Fig. 13).

Rice. 13. Triangle

It has three angles - angle, angle and angle. The points , , are called the vertices of the triangle. Three segments - the segment , , are the sides of the triangle.

Let's repeat what types of triangles are distinguished (see Fig. 14).

Rice. 14. Types of triangles

According to the types of angles, triangles can be divided into acute-angled, right-angled and obtuse-angled triangles. In a triangle, all angles are acute, such a triangle is called an acute triangle. A triangle has a right angle, such a triangle is called a right triangle. A triangle has an obtuse angle, such a rectangle is called an obtuse triangle.

By whether the lengths of the sides are equal, triangles are distinguished:

Versatile - such triangles have different lengths of all sides;

Equilateral - these triangles have the same lengths of all sides;

Isosceles - they have the same length of the two sides. Two sides of equal length are called the sides of the triangle, and the third side is the base of the triangle (see Fig. 15).

Rice. 15. Types of triangles

What shapes are called polygons? If you connect several points in series so that their connection gives a closed broken line, then an image of a polygon, quadrangle, five- or hexagon, etc. is created.

Polygons are named according to the number of angles. Each polygon has as many vertices and sides as it has corners (see Figure 16).

Rice. 16. Polygons

All the figures depicted (see Fig. 17) are called quadrilaterals. Why?

Rice. 17. Quadrangles

You probably noticed that all figures have four corners, but they can all be divided into two groups. How would you do it?

Probably, you singled out quadrangles in a separate group, in which all corners are right, and such quadrangles were called rectangular quadrangles. The opposite sides of the rectangles are equal (see Fig. 18).

Rice. 18. Rectangular quadrilaterals

In a rectangle, and are opposite sides, and they are equal, and are also opposite sides, and they are equal (see Fig. 19).

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Introduction

Geometry is one of the most important components of mathematical education, necessary for the acquisition of specific knowledge about space and practically significant skills, the formation of a language for describing objects of the surrounding world, for the development of spatial imagination and intuition, mathematical culture, as well as for aesthetic education. The study of geometry contributes to the development of logical thinking, the formation of proof skills.

The 7th grade geometry course systematizes knowledge about the simplest geometric shapes and their properties; the concept of equality of figures is introduced; the ability to prove the equality of triangles with the help of the studied signs is developed; a class of construction problems with the help of a compass and straightedge is introduced; one of the most important concepts is introduced - the concept of parallel lines; new interesting and important properties of triangles are considered; one of the most important theorems in geometry is considered - the theorem on the sum of angles of a triangle, which allows us to give a classification of triangles by angles (acute-angled, rectangular, obtuse-angled).

During classes, especially when moving from one part of the lesson to another, changing activities, the question arises of maintaining interest in classes. In this way, relevant the question arises of applying tasks in the classroom in geometry, in which there is a condition of the problem situation and elements of creativity. In this way, goal of this study is the systematization of tasks of geometric content with elements of creativity and problem situations.

Object of study: Problems in geometry with elements of creativity, entertainment and problem situations.

Research objectives: To analyze the existing problems in geometry, aimed at the development of logic, imagination and creative thinking. Show how entertaining techniques can develop interest in the subject.

Theoretical and practical significance of the research consists in the fact that the collected material can be used in the process of additional classes in geometry, namely at olympiads and competitions in geometry.

Scope and structure of the study:

The study consists of an introduction, two chapters, a conclusion, a bibliographic list, contains 14 pages of the main typewritten text, 1 table, 10 figures.

Chapter 1. FLAT GEOMETRIC FIGURES. BASIC CONCEPTS AND DEFINITIONS

1.1. Basic geometric shapes in the architecture of buildings and structures

In the world around us, there are many material objects of various shapes and sizes: residential buildings, machine parts, books, jewelry, toys, etc.

In geometry, instead of the word object, they say a geometric figure, while dividing geometric figures into flat and spatial ones. In this paper, one of the most interesting sections of geometry will be considered - planimetry, in which only plane figures are considered. Planimetry(from Latin planum - “plane”, other Greek μετρεω - “I measure”) - a section of Euclidean geometry that studies two-dimensional (single-plane) figures, that is, figures that can be placed within the same plane. A flat geometric figure is one whose all points lie on the same plane. An idea of such a figure is given by any drawing made on a piece of paper.

But before considering flat figures, it is necessary to get acquainted with simple, but very important figures, without which flat figures simply cannot exist.

The simplest geometric figure is dot. This is one of the main figures of geometry. It is very small, but it is always used to build various forms on a plane. The point is the main figure for absolutely all constructions, even the highest complexity. From the point of view of mathematics, a point is an abstract spatial object that does not have such characteristics as area, volume, but at the same time remains a fundamental concept in geometry.

Straight- one of the fundamental concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined by the axioms of geometry (Euclidean). If the basis for constructing geometry is the concept of the distance between two points in space, then a straight line can be defined as a line along which the path along which is equal to the distance between two points.

Straight lines in space can occupy different positions, we will consider some of them and give examples that are found in the architectural appearance of buildings and structures (Table 1):

Table 1

Parallel lines	Properties of parallel lines
	If the lines are parallel, then their projections of the same name are parallel:	Essentuki, the building of the mud baths (author's photo)
intersecting lines	Properties of intersecting lines	Examples in the architecture of buildings and structures
	Intersecting lines have a common point, that is, the points of intersection of their projections of the same name lie on a common line of communication:	Mountain buildings in Taiwan https://www.sro-ps.ru/novosti_otrasli/2015_11_11_pervoe_zdanie_iz_grandioznogo_proekta_big_v_tayvane
Crossed lines	Properties of skew lines	Examples in the architecture of buildings and structures
	Straight lines that do not lie in the same plane and are not parallel to each other are intersecting. None is a common line of communication. If intersecting and parallel lines lie in the same plane, then skew lines lie in two parallel planes.	Robert, Hubert Villa Madama near Rome https://gallerix.ru/album/Hermitage-10/pic/glrx-172894287

1.2. Flat geometric figures. Properties and definitions

Observing the forms of plants and animals, mountains and meanders of rivers, the peculiarities of the landscape and distant planets, man borrowed from nature its correct forms, sizes and properties. Material needs prompted a person to build dwellings, make tools for labor and hunting, sculpt dishes from clay, and so on. All this gradually contributed to the fact that a person came to the realization of the basic geometric concepts.

Quadrangles:

Parallelogram(ancient Greek παραλληλόγραμμον from παράλληλος - parallel and γραμμή - line, line) is a quadrangle whose opposite sides are pairwise parallel, that is, they lie on parallel lines.

Features of a parallelogram:

A quadrilateral is a parallelogram if one of the following conditions is satisfied: 1. If opposite sides in a quadrilateral are pairwise equal, then the quadrilateral is a parallelogram. 2. If in a quadrilateral the diagonals intersect and the intersection point is divided in half, then this quadrilateral is a parallelogram. 3. If in a quadrilateral two sides are equal and parallel, then this quadrilateral is a parallelogram.

A parallelogram with all right angles is called rectangle.

A parallelogram with all sides equal is called rhombus.

Trapeze— is a quadrilateral in which two sides are parallel and the other two sides are not parallel. Also, a quadrilateral is called a trapezoid, in which one pair of opposite sides is parallel, and the sides are not equal to each other.

Triangle- This is the simplest geometric figure formed by three segments that connect three points that do not lie on one straight line. These three points are called vertices. triangle, and the segments are sides triangle. It is because of its simplicity that the triangle was the basis of many measurements. Land surveyors in their calculations of land areas and astronomers in finding the distances to planets and stars use the properties of triangles. This is how the science of trigonometry arose - the science of measuring triangles, of expressing sides through its angles. The area of any polygon is expressed in terms of the area of a triangle: it is enough to divide this polygon into triangles, calculate their areas and add the results. True, it was not immediately possible to find the correct formula for the area of \u200b\u200ba triangle.

The properties of the triangle were especially actively studied in the 15th-16th centuries. Here is one of the most beautiful theorems of that time, due to Leonhard Euler:

A huge amount of work on the geometry of the triangle, carried out in the XY-XIX centuries, created the impression that everything is already known about the triangle.

Polygon - it is a geometric figure, usually defined as a closed polyline.

A circle- the locus of points in the plane, the distance from which to a given point, called the center of the circle, does not exceed a given non-negative number, called the radius of this circle. If the radius is zero, then the circle degenerates into a point.

There are a large number of geometric shapes, they all differ in parameters and properties, sometimes surprising with their shapes.

In order to better remember and distinguish flat figures by properties and features, I came up with a geometric fairy tale, which I would like to bring to your attention in the next paragraph.

Chapter 2

2.1. Puzzles for building a complex figure from a set of flat geometric elements.

Having studied flat figures, I thought, are there any interesting problems with flat figures that can be used as tasks-games or tasks-puzzles. And the first problem I found was the Tangram puzzle.

This is a Chinese puzzle. In China, it is called "chi tao tu", i.e. a seven-piece mental puzzle. In Europe, the name "Tangram" most likely arose from the word "tan", which means "Chinese" and the root "gram" (Greek - "letter").

First you need to draw a square 10 x10 and divide it into seven parts: five triangles 1-5 , square 6 and parallelogram 7 . The essence of the puzzle is to use all seven pieces to put together the figures shown in Figure 3.

Fig.3. Elements of the game "Tangram" and geometric shapes

Fig.4. Tasks "Tangram"

It is especially interesting to make “figurative” polygons from flat figures, knowing only the outlines of objects (Fig. 4). I came up with several of these outline tasks myself and showed these tasks to my classmates, who gladly began to solve the tasks and made up many interesting polyhedral figures similar to the outlines of objects in the world around us.

To develop the imagination, you can also use such forms of entertaining puzzles as tasks for cutting and reproducing given shapes.

Example 2. Cutting (parquet) problems may seem, at first glance, to be very diverse. However, most of them use only a few basic types of cuts (usually those that can be used to get another from one parallelogram).

Let's take a look at some cutting techniques. In this case, the cut figures will be called polygons.

Rice. 5. Cutting techniques

Figure 5 shows geometric shapes from which you can assemble various ornamental compositions and make an ornament with your own hands.

Example 3. Another interesting task that you can come up with and share with other students, while whoever collects the most cut pieces is declared the winner. There can be quite a few tasks of this type. For coding, you can take all existing geometric shapes that are cut into three or four parts.

Fig.6. Examples of tasks for cutting:

------ - recreated square; - cut with scissors;

Main figure

2.2 Equal-sized and equally composed figures

Consider another interesting technique for cutting flat figures, where the main "heroes" of cutting will be polygons. When calculating the areas of polygons, a simple trick called the partitioning method is used.

In general, polygons are said to be equally composed if, after cutting the polygon in a certain way F into a finite number of parts, it is possible, by arranging these parts differently, to form a polygon H out of them.

From this follows the following theorem: Equally composed polygons have the same area, so they will be considered equal areas.

Using the example of equally composed polygons, one can also consider such an interesting cutting as the transformation of the "Greek cross" into a square (Fig. 7).

Fig.7. Transformation of the "Greek cross"

In the case of a mosaic (parquet) made up of Greek crosses, the period parallelogram is a square. We can solve the problem by overlaying a tiling of squares onto a tiling of crosses so that the congruent points of one tiling coincide with the congruent points of the other (Fig. 8).

In the figure, the congruent points of the mosaic of crosses, namely the centers of the crosses, coincide with the congruent points of the "square" mosaic - the vertices of the squares. By shifting the square tiling in parallel, we always get a solution to the problem. Moreover, the task has several solutions, if color is used in the preparation of the parquet ornament.

Fig.8. Parquet assembled from a Greek cross

Another example of equally composed figures can be considered on the example of a parallelogram. For example, a parallelogram is equidistant with a rectangle (Fig. 9).

This example illustrates the method of partitioning, which consists in the fact that in order to calculate the area of a polygon, one tries to divide it into a finite number of parts in such a way that from these parts it is possible to form a simpler polygon, the area of which we already know.

For example, a triangle is equidistant with a parallelogram having the same base and half the height. From this position, the formula for the area of a triangle is easily derived.

Note that for the above theorem, we also have converse theorem: if two polygons are equal in size, then they are equal.

This theorem, proved in the first half of the XIX century. by the Hungarian mathematician F. Bolyai and the German officer and mathematician P. Gervin, can also be represented in this form: if there is a cake in the shape of a polygon and a polygonal box of a completely different shape, but the same area, then you can cut the cake into a finite number of pieces (without turning them cream down) that they can be put into this box.

Conclusion

In conclusion, I note that problems for flat figures are sufficiently represented in various sources, but I was interested in those on the basis of which I had to come up with my own puzzle problems.

After all, solving such problems, you can not only accumulate life experience, but also acquire new knowledge and skills.

In puzzles, when building actions-moves using rotations, shifts, transfers on planes or their compositions, I got new images created by myself, for example, polyhedron figures from the Tangram game.

It is known that the main criterion for the mobility of a person's thinking is the ability to perform certain actions in a set period of time, and in our case, moves of figures on a plane, by means of recreating and creative imagination. Therefore, studying mathematics and, in particular, geometry at school will give me even more knowledge in order to further apply them in my future professional activities.

Bibliographic list

1. Pavlova, L.V. Non-traditional approaches to teaching drawing: textbook / L.V. Pavlova. - Nizhny Novgorod: Publishing house of NSTU, 2002. - 73 p.

2. Encyclopedic dictionary of a young mathematician / Comp. A.P. Savin. - M.: Pedagogy, 1985. - 352 p.

3.https://www.srops.ru/novosti_otrasli/2015_11_11_pervoe_zdanie_iz_grandioznogo_proekta_big_v_tayvane

4.https://www.votpusk.ru/country/dostoprim_info.asp?ID=16053

Attachment 1

Questionnaire for classmates

1. Do you know what a Tangram puzzle is?

2. What is a "Greek cross"?

3. Would you be interested to know what "Tangram" is?

4. Would you be interested to know what a "Greek cross" is?

22 students of the 8th grade were interviewed. Results: 22 students do not know what "Tangram" and "Greek cross" are. 20 students would be interested to know how to get a more complex figure using the Tangram puzzle, consisting of seven flat figures. The results of the survey are summarized in the diagram.

Appendix 2

Elements of the game "Tangram" and geometric shapes

Transformation of the "Greek cross"

Geometry is a branch of mathematics that studies shapes and their properties.

The geometry that is studied at school is called Euclidean, after the ancient Greek scientist Euclid (3rd century BC).

The study of geometry begins with planimetry. Planimetry- This is a branch of geometry in which figures are studied, all parts of which are in the same plane.

Geometric figures

In the world around us, there are many material objects of various shapes and sizes: residential buildings, machine parts, books, jewelry, toys, etc.

In geometry, instead of the word object, they say a geometric figure. Geometric figure(or short: figure) is a mental image of a real object, in which only the shape and dimensions are stored, and only they are taken into account.

Geometric shapes are divided into flat and spatial. In planimetry, only plane figures are considered. A flat geometric figure is one whose all points lie on the same plane. An idea of such a figure is given by any drawing made on a piece of paper.

Geometric shapes are very diverse, for example, a triangle, a square, a circle, etc.:

A part of any geometric figure (except for a point) is also a geometric figure. The union of several geometric shapes will also be a geometric figure. In the figure below, the left figure is made up of a square and four triangles, while the right figure is made up of a circle and parts of a circle.

Geometric figure- a set of points on a surface (often on a plane), which forms a finite number of lines.

The main geometric figures on the plane are dot and straight line. A segment, a ray, a broken line are the simplest geometric figures on a plane.

Dot- the smallest geometric figure, which is the basis of other figures in any image or drawing.

Each more complex geometric figure there is a set of points that have a certain property, characteristic only for this figure.

Straight line, or straight - this is an infinite set of points located on the 1st line, which has no beginning and end. On a sheet of paper, you can see only part of a straight line, because. it has no limit.

The line is drawn like this:

The part of a straight line that is bounded on 2 sides by points is called segment straight or cut. He is portrayed like this:

Ray is a directed half-line that has a point of origin and which has no end. The beam is shown like this:

If you put a point on a straight line, then this point will split the straight line into 2 oppositely directed beams. These rays are called additional.

broken line- several segments that are connected to each other in such a way that the end of the 1st segment is the beginning of the 2nd segment, and the end of the 2nd segment is the beginning of the 3rd segment, and so on, with neighboring (which have 1-well in common point) the segments are located on different straight lines. When the end of the last segment does not coincide with the beginning of the 1st, then this broken line will be called open:

When the end of the last segment of the polyline coincides with the beginning of the 1st, then this polyline will be closed. An example of a closed polyline is any polygon:

Four-link closed polyline - quadrilateral (rectangle):