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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 in 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 and the formation of proof skills.
The 7th grade geometry course systematizes knowledge about the simplest geometric figures and their properties; the concept of equality of figures is introduced; the ability to prove the equality of triangles using the studied signs is developed; a class of problems involving construction using a compass and ruler 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 the angles of a triangle, which allows us to classify triangles by angles (acute, rectangular, obtuse).
During classes, especially when moving from one part of the lesson to another, changing activities, the question arises of maintaining interest in classes. Thus, relevant The question arises about using tasks in geometry classes that involve the condition of a problem situation and elements of creativity. Thus, purpose This study is to systematize tasks of geometric content with elements of creativity and problem situations.
Object of study: Geometry tasks with elements of creativity, entertainment and problem situations.
Research objectives: Analyze existing geometry tasks aimed at developing logic, imagination and creative thinking. Show how you can develop interest in a subject using entertaining techniques.
Theoretical and practical significance of the research is that the collected material can be used in the process of additional lessons 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 bibliography, contains 14 pages of main typewritten text, 1 table, 10 figures.
Chapter 1. FLAT GEOMETRIC FIGURES. BASIC CONCEPTS AND DEFINITIONS
1.1. Basic geometric figures in the architecture of buildings and structures
In the world around us, there are many material objects of different shapes and sizes: residential buildings, machine parts, books, jewelry, toys, etc.
In geometry, instead of the word object, they say geometric figure, while dividing geometric figures into flat and spatial. In this work, we will consider one of the most interesting sections of geometry - planimetry, in which only plane figures are considered. Planimetry(from Latin planum - “plane”, ancient Greek μετρεω - “measure”) - a section of Euclidean geometry that studies two-dimensional (single-plane) figures, that is, figures that can be located within the same plane. A flat geometric figure is one in which all points lie on the same plane. Any drawing made on a sheet of paper gives an idea of such a figure.
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 shapes on a plane. The point is the main figure for absolutely all constructions, even the highest complexity. From a mathematical point of view, a point is an abstract spatial object that does not have such characteristics as area or 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 distance between two points in space, then a straight line can be defined as a line along which the path is equal to the distance between two points.
Straight lines in space can occupy different positions; let’s consider some of them and give examples found in the architectural appearance of buildings and structures (Table 1):
Table 1
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Parallel lines |
Properties of parallel lines |
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If the lines are parallel, then their projections of the same name are parallel: |
Essentuki, mud bath building (photo by the author) |
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Intersecting lines |
Properties of intersecting lines |
Examples in the architecture of buildings and structures |
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Intersecting lines have a common point, that is, the intersection points of their projections of the same name lie on a common connection line: |
"Mountain" buildings in Taiwan https://www.sro-ps.ru/novosti_otrasli/2015_11_11_pervoe_zdanie_iz_grandioznogo_proekta_big_v_tayvane |
|
|
Crossing lines |
Properties of skew lines |
Examples in the architecture of buildings and structures |
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Straight lines that do not lie in the same plane and are not parallel to each other are intersecting. None is a common communication line. If intersecting and parallel lines lie in the same plane, then intersecting 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 shapes. Properties and Definitions
Observing the forms of plants and animals, mountains and river meanders, landscape features and distant planets, man borrowed from nature its correct forms, sizes and properties. Material needs prompted people to build houses, make tools for labor and hunting, sculpt dishes from clay, and so on. All this gradually contributed to the fact that man came to understand the basic geometric concepts.
Quadrilaterals:
Parallelogram(ancient Greek παραλληλόγραμμον from παράλληλος - parallel and γραμμή - line, line) is a quadrilateral whose opposite sides are pairwise parallel, that is, they lie on parallel lines.
Signs of a parallelogram:
A quadrilateral is a parallelogram if one of the following conditions is met: 1. If in a quadrilateral the opposite sides are pairwise equal, then the quadrilateral is a parallelogram. 2. If in a quadrilateral the diagonals intersect and are divided in half by the point of intersection, then this quadrilateral is a parallelogram. 3. If two sides of a quadrilateral are equal and parallel, then this quadrilateral is a parallelogram.
A parallelogram whose angles are all right angles is called rectangle.
A parallelogram in which all sides are equal is called diamond
Trapezoid— It is a quadrilateral in which two sides are parallel and the other two sides are not parallel. Also, a trapezoid is a quadrilateral in which one pair of opposite sides is parallel, and the sides are not equal to each other.
Triangle is the simplest geometric figure formed by three segments that connect three points that do not lie on the same straight line. These three points are called vertices triangle, and the segments are sides triangle. It is precisely because of its simplicity that the triangle was the basis of many measurements. Land surveyors, when calculating land areas, and astronomers, when finding 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 the sides through its angles. The area of any polygon is expressed through 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 a 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 was already known about the triangle.
Polygon - it is a geometric figure, usually defined as a closed polyline.
Circle- the geometric 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 characteristics, I came up with a geometric fairy tale, which I would like to present to your attention in the next paragraph.
Chapter 2. PUZZLES FROM FLAT GEOMETRIC FIGURES
2.1.Puzzles for constructing a complex figure from a set of flat geometric elements.
After studying flat shapes, I wondered if there were any interesting problems with flat shapes that could be used as games or puzzles. And the first problem I found was the Tangram puzzle.
This is a Chinese puzzle. In China it is called "chi tao tu", or 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 10 x 10 square 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 Fig. 3.
Fig.3. Elements of the game "Tangram" and geometric shapes
Fig.4. Tangram tasks
It is especially interesting to make “shaped” 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 happily began to solve the tasks and created many interesting polyhedral figures, similar to the outlines of objects in the world around us.
To develop imagination, you can also use such forms of entertaining puzzles as tasks for cutting and reproducing given figures.
Example 2. Cutting (parqueting) tasks may seem, at first glance, to be quite diverse. However, most of them use only a few basic types of cuts (usually those that can be used to create another from one parallelogram).
Let's look at some cutting techniques. In this case, we will call the cut figures polygons.
Rice. 5. Cutting techniques
Figure 5 shows geometric shapes from which you can assemble various ornamental compositions and create an ornament with your own hands.
Example 3. Another interesting task that you can come up with on your own and exchange with other students, and whoever collects the most cut pieces is declared the winner. There can be quite a lot of tasks of this type. For coding, you can take all existing geometric shapes, which are cut into three or four parts.
Fig. 6. Examples of cutting tasks:
------ - recreated square; - cut with scissors;
Basic figure
2.2. Equal-sized and equally-composed figures
Let's consider another interesting technique for cutting flat figures, where the main “heroes” of the cuts will be polygons. When calculating the areas of polygons, a simple technique called the partitioning method is used.
In general, polygons are called equiconstituted 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 from them.
This leads to the following theorem: Equilateral polygons have the same area, so they will be considered equal in area.
Using the example of equipartite polygons, we can consider such an interesting cutting as the transformation of a “Greek cross” into a square (Fig. 7).
Fig.7. Transformation of the "Greek Cross"
In the case of a mosaic (parquet) composed of Greek crosses, the parallelogram of the periods is a square. We can solve the problem by superimposing a mosaic made of squares onto a mosaic formed with the help of crosses, so that the congruent points of one mosaic 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 moving the square mosaic in parallel, we will always obtain a solution to the problem. Moreover, the problem has several possible solutions if color is used when composing the parquet ornament.
Fig.8. Parquet made from a Greek cross
Another example of equally proportioned figures can be considered using the example of a parallelogram. For example, a parallelogram is equivalent to a rectangle (Fig. 9).
This example illustrates the partitioning method, which consists in calculating the area of a polygon by trying to divide it into a finite number of parts in such a way that these parts can be used to create a simpler polygon whose area we already know.
For example, a triangle is equivalent to 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 the above theorem also holds converse theorem: if two polygons are equal in size, then they are equivalent.
This theorem, proven in the first half of the 19th century. by the Hungarian mathematician F. Bolyai and the German officer and mathematics lover P. Gerwin, can be represented in this way: 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 side down) that they can be placed in this box.
Conclusion
In conclusion, I would like to note that there are quite a lot of problems on flat figures in various sources, but those that were of interest to me were the ones on the basis of which I had to come up with my own puzzle problems.
After all, by solving such problems, you can not only accumulate life experience, but also acquire new knowledge and skills.
In puzzles, when constructing actions-moves using rotations, shifts, translations on a plane or their compositions, I got independently created new images, for example, polyhedron figures from the game “Tangram”.
It is known that the main criterion for the mobility of a person’s thinking is the ability, through reconstructive and creative imagination, to perform certain actions within a set period of time, and in our case, moves of figures on a plane. Therefore, studying mathematics and, in particular, geometry at school will give me even more knowledge to later apply in my future professional activities.
Bibliography
1. Pavlova, L.V. Non-traditional approaches to teaching drawing: a textbook / L.V. Pavlova. - Nizhny Novgorod: NSTU Publishing House, 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
Annex 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 8th grade students were surveyed. Results: 22 students do not know what “Tangram” and “Greek cross” are. 20 students would be interested in learning how to use the Tangram puzzle, consisting of seven flat figures, to obtain a more complex figure. The survey results are summarized in a diagram.
Appendix 2
Elements of the game "Tangram" and geometric shapes
Transformation of the "Greek Cross"
1. The concept of a geometric figure.
3. Parallel and perpendicular lines.
4. Triangles.
5. Quadrilaterals.
6. Polygons.
7. Circle and circle.
8. Construction of geometric figures on a plane.
9. Transformations of geometric shapes. Transformation concept
Main literature;
additional literature
Geometric figure concept
Geometric figure defined as any set of points.
Segment, straight line, circle, ball- geometric figures.
If all points of a geometric figure belong to one 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 contained in another), we can consider the union, intersection and difference of figures.
For example, combining two rays AB And MK(Fig. 1) is straight KV, and their intersection is a segment AM.
K A M V
Convex figures are a plane, a straight line, a ray, a segment, and a point. It is easy to verify that the convex figure is a circle (Fig. 3). If we continue the XY segment until it intersects with the circle, we get a chord AB. Since the chord is contained in a circle, the segment XY is also contained in the circle and, therefore, the circle is a convex figure.
For polygons, another definition is known: a polygon is called convex if it lies on one side of each line containing its side .
Since the equivalence of this definition and the one given above for a polygon has been proven, we can use both.
Based on these concepts, let's consider other geometric figures studied in the school planimetry course. Let us consider their definitions and basic properties, accepting them without proof. Knowledge of this material and the ability to apply it to solving simple geometric problems is the basis on which it is possible to build a methodology for teaching elementary schoolchildren the elements of geometry.
Angles
Let us remind you that an angle 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 designated 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: Ð A, Ð (k, l), Ð ABC.
The angle is called expanded , if its sides lie on the same straight line.
An angle that is half a straight angle is called direct. An angle less than a right angle is called spicy. An angle greater than a right angle but less than a straight angle is called stupid .
In addition to the concept of an angle given above, in geometry the concept of a plane angle is considered.
A plane angle is a part of a plane bounded by two different rays emanating from the same point.
The angles considered in planimetry do not exceed the unfolded angle.
The two angles are called adjacent, if they have one side in common, and the other sides of these angles are additional half-lines.
The sum of adjacent angles is 180°. The validity of this property follows from the definition of adjacent angles.
The two angles are called vertical, if the sides of one angle are complementary half-lines of the sides of the other. Angles AOB and COB, as well as angles AOC and D0B, are vertical (Fig. 4).
A segment is denoted in the same way as a straight line. A segment is a part of a line along with the points limiting this part. It is clear that two points should not coincide, that is, lie in the same place on a straight line. If you put a point on a straight line, then with this point the straight line is divided into two rays, oppositely directed. Points are designated in capital Latin letters, lines are designated in small Latin letters. That a straight line passes through these two points, and only one. This seems to be understandable.
A plane, like a straight line, cannot see either the beginning or the end. We consider only the part of the plane that is limited by a closed polyline. A segment, a ray, a broken line are the simplest geometric shapes on a plane. A point is the smallest geometric figure that is the basis of other figures in any image or drawing.
Usually, for a straight line segment, it does not matter in what order its ends are considered: that is, segments AB(\displaystyle AB) and BA(\displaystyle BA) represent the same segment. For example, the directed segments AB(\displaystyle AB) and BA(\displaystyle BA) do not coincide. Further generalization leads to the concept of a vector - the class of all equal in length and codirectional directed segments.
A ray starting at point O and containing point A is called "ray OA". You left the apartment, bought bread at the store, went into the entrance and started talking with your neighbor. What line did you get? Problem: where is the line, ray, segment, curve?
The links of a broken line (similar to the links of a chain) are the segments that make up the broken line. Adjacent links are links in which the end of one link is the beginning of another. Adjacent links should not lie on the same straight line. Adjacent vertices are the endpoints of one side of the polygon. My son goes to school to prepare. Given in the book “One-step, Two-step...” (Peterson and Kholina) the task “Find straight lines, rays and segments.”
The straight line is one of the fundamental concepts of geometry. However, we can say that this is a geometric figure that is obtained from a segment by unlimitedly extending it in both directions. A curve or line is a geometric concept, defined differently in different branches of geometry, sometimes defined as “length without width” or as “the boundary of a figure.”
Kandinsky systematized his views on painting in the book “Point and Line on a Plane” (1926). The variety of lines depends on the number of these forces and their combinations. In the end, all forms of lines can be reduced to two cases: 1.
So, the horizontal is a cold load-bearing base that can be extended on a plane in various directions. Coldness and flatness are the main sounds of this line, it can be defined as the shortest form of unlimited cold possibility of movement.2. Completely opposite to this line, both externally and internally, is a vertical standing at right angles to it, in which flatness is replaced by height, that is, cold by heat.
Even among the simplest figures, the simplest one stands out - this is the point. All other figures consist of many points. In geometry, it is customary to denote points in capital (capital) Latin letters. A straight line is an infinite line on which if you take any two points, the shortest distance between them will be along this straight line.
For example, straight line a, straight line b. However, in some cases there are two big ones. Otherwise, the segment will have zero length and will essentially be a point. Segments are designated by two capital letters, which indicate the ends of the segment.
Basic Geometric Concepts
Thus, if a segment is limited at both ends, then the ray has only one side, and the other side of the ray is infinite, like a straight line. Rays are designated in the same way as straight lines: either with one small letter or two large ones.
In geometry there is a section that deals with the study of various figures on a plane and is called planimetry. You already know that a figure is an arbitrary set of points located on a plane. From the material studied above, you already know that the point refers to the main geometric figures. After all, the construction of more complex geometric figures consists of many points characteristic of a given figure.
A figure that has two rays and a vertex is called an angle. The junction of the rays is the vertex of this angle, and its sides are the rays that form this angle. The triangle you have already studied also belongs to simple geometric figures. This is one of the types of polygons in which part of the plane is limited by three points and three segments that connect these points in pairs.
In a polygon, all points that connect the segments are its vertices. And the segments that make up a polygon are its sides. But one of the famous paintings, created at the beginning of the last century by Malevich, glorifies such a geometric figure as a square.
In the future there will be definitions for different figures except two - a point and a straight line. This means that sometimes we can denote a straight line with two capital Latin letters, for example, straight line \(AB\), since no other straight line can be drawn through these two points. 2) All lines \(a\), \(b\) and \(c\) intersect! This is the study of figures, their properties and relative positions. The first geometric facts were found in Babylonian cuneiform tables and Egyptian papyri (3rd millennium BC), as well as in other sources.
A point is the smallest geometric figure, which is the basis of all other constructions (figures) in any image or drawing. The part of a line bounded by two points and a point is called a segment. A plane, like a straight line, is an initial concept that has no definition.
Geometric figures are a complex of points, lines, solids or surfaces. These elements can be located both on the plane and in space, forming a finite number of straight lines.
The term “figure” implies several sets of points. They must be located on one or more planes and at the same time limited to a specific number of completed lines.
The main geometric figures are the point and the straight line. They are located on a plane. In addition to them, among simple figures there is a ray, a broken line and a segment.
Dot
This is one of the main figures of geometry. It is very small, but it is always used to build various shapes on a plane. The point is the main figure for absolutely all constructions, even the highest complexity. In geometry, it is usually denoted by a letter of the Latin alphabet, for example, A, B, K, L.
From a mathematical point of view, a point is an abstract spatial object that does not have such characteristics as area or volume, but at the same time remains a fundamental concept in geometry. This zero-dimensional object simply has no definition.
Straight
This figure is completely placed in one plane. A straight line does not have a specific mathematical definition, since it consists of a huge number of points located on one endless line, which has no limit or boundaries.
There is also a segment. This is also a straight line, but it starts and ends from a point, which means it has geometric limitations.
The line can also turn into a directional beam. This happens when a straight line starts from a point, but does not have a clear ending. If you put a point in the middle of the line, then it will split into two rays (additional), and oppositely directed to each other.
Several segments that are sequentially connected to each other by ends at a common point and are not located on the same straight line are usually called a broken line.
Corner
Geometric figures, the names of which we discussed above, are considered key elements used in the construction of more complex models.
An angle is a structure consisting of a vertex and two rays that extend from it. That is, the sides of this figure connect at one point.
Plane
Let's consider another primary concept. A plane is a figure that has neither end nor beginning, as well as a straight line and a point. When considering this geometric element, only its part, limited by the contours of a broken closed line, is taken into account.
Any smooth bounded surface can be considered a plane. This could be an ironing board, a piece of paper, or even a door.
Quadrilaterals
A parallelogram is a geometric figure whose opposite sides are parallel to each other in pairs. Among the particular types of this design are diamond, rectangle and square.
A rectangle is a parallelogram in which all sides touch at right angles.
A square is a quadrilateral with equal sides and angles.
A rhombus is a figure in which all sides are equal. In this case, the angles can be completely different, but in pairs. Each square is considered a diamond. But in the opposite direction this rule does not always apply. Not every rhombus is a square.
Trapezoid
Geometric shapes can be completely different and bizarre. Each of them has a unique shape and properties.
A trapezoid is a figure that is somewhat similar to a quadrilateral. It has two parallel opposite sides and is considered curved.
Circle
This geometric figure implies the location on one plane of points equidistant from its center. In this case, a given non-zero segment is usually called a radius.
Triangle
This is a simple geometric figure that is very often encountered and studied.
A triangle is considered a subtype of a polygon, located on one plane and limited by three edges and three points of contact. These elements are connected in pairs.
Polygon
The vertices of polygons are the points connecting the segments. And the latter, in turn, are considered to be parties.
Volumetric geometric shapes
- prism;
- sphere;
- cone;
- cylinder;
- pyramid;
These bodies have something in common. All of them are limited to a closed surface, inside of which there are many points.
Volumetric bodies are studied not only in geometry, but also in crystallography.
Curious facts
Surely you will be interested in reading the information provided below.
- Geometry was formed as a science back in ancient times. This phenomenon is usually associated with the development of art and various crafts. And the names of geometric figures indicate the use of the principles of determining similarity and similarity.
- Translated from ancient Greek, the term “trapezoid” means a table for a meal.
- If you take different shapes whose perimeter is the same, then the circle is guaranteed to have the largest area.
- Translated from Greek, the term “cone” means a pine cone.
- There is a famous painting by Kazemir Malevich, which, since the last century, has attracted the views of many painters. The work “Black Square” has always been mystical and mysterious. The geometric figure on the white canvas delights and amazes at the same time.
There are a large number of geometric shapes. They all differ in parameters, and sometimes even surprise in shape.
2.1. Geometric shapes on a plane
In recent years, there has been a tendency to include a significant amount of geometric material in the initial mathematics course. But in order to introduce students to various geometric figures and teach them how to depict correctly, he needs appropriate mathematical training. The teacher must be familiar with the leading ideas of the geometry course, know the basic properties of geometric figures, and be able to construct them.
When depicting a flat figure, no geometric problems arise. The drawing serves either as an exact copy of the original or represents a similar figure to it. Looking at the image of a circle in the drawing, we get the same visual impression as if we were looking at the original circle.
Therefore, the study of geometry begins with planimetry.
Planimetry is a branch of geometry in which figures on a plane are studied.
A geometric figure is defined as any set of points.
A segment, a straight line, a circle are geometric shapes.
If all the points of a geometric figure belong to one 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; we can consider the union, intersection and difference of figures.
For example, the union of two rays AB and MK is the straight line KB, and their intersection is the segment AM.
There are convex and non-convex figures. A figure is called convex if, together with any two of its points, it also contains a segment connecting them.
Figure F 1 is convex, and figure F 2 is non-convex.
Convex figures are a plane, a straight line, a ray, a segment, and a point. It is not difficult to verify that the convex figure is a circle.
If we continue the segment XY until it intersects with the circle, we get the chord AB. Since the chord is contained in a circle, the segment XY is also contained in the circle, and, therefore, the circle is a convex figure.
The basic properties of the simplest figures on the plane are expressed in the following axioms:
1. Whatever the line, there are points that belong to this line and do not belong to it.
Through any two points you can draw a straight line, and only one.
This axiom expresses the basic property of belonging to points and lines on the plane.
2. Of the three points on a line, one and only one lies between the other two.
This axiom expresses the basic property of the location of points on a straight line.
3. Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.
Obviously, axiom 3 expresses the main property of measuring segments.
This sentence expresses the basic property of the location of points relative to a straight line on a plane.
5. Each angle has a certain degree measure greater than zero. The unfolded angle is 180°. The degree measure of an angle is equal to the sum of the degree measures of the angles into which it is divided by any ray passing between its sides.
This axiom expresses the basic property of measuring angles.
6. On any half-line from its starting point, you can plot a segment of a given length, and only one.
7. From any half-line, into a given half-plane, you can put an angle with a given degree measure less than 180 O, and only one.
These axioms reflect the basic properties of laying out angles and segments.
The basic properties of the simplest figures include the existence of a triangle equal to the given one.
8. Whatever the triangle, there is an equal triangle in a given location relative to a given half-line.
The basic properties of parallel lines are expressed by the following axiom.
9. Through a point not lying on a given line, no more than one straight line parallel to the given one can be drawn on the plane.
Let's look at some geometric shapes that are studied in elementary school.
An angle 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 called developed 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 acute. An angle greater than a right angle but less than a straight angle is called an obtuse angle.
In addition to the concept of an angle given above, in geometry the concept of a plane angle is considered.
A plane angle is a part of a plane bounded by two different rays emanating from one point.
There are two plane angles formed by two rays with a common origin. They are called additional. The figure shows two plane angles with sides OA and OB, one of them is shaded.
Angles can be adjacent or vertical.
Two angles are called adjacent if they have one side in common, and the other sides of these angles are complementary half-lines.
The sum of adjacent angles is 180 degrees.
Two angles are called vertical if the sides of one angle are complementary half-lines of the sides of the other.
Angles AOD and SOV, as well as angles AOS and DOV are vertical.
Vertical angles are equal.
Parallel and perpendicular lines.
Two lines in a plane are called parallel if they do not intersect.
If line a is parallel to line b, then write a II c.
Two lines are called perpendicular if they intersect at right angles.
If line a is perpendicular to line b, then write a b.
Triangles.
A triangle is a geometric figure that consists of three points that do not lie on the same line and three pairwise segments connecting them.
Any triangle divides the plane into two parts: internal and external.
In any triangle, the following elements are distinguished: sides, angles, altitudes, bisectors, medians, midlines.
The altitude of a triangle dropped from a given vertex is the perpendicular drawn from this vertex to the line containing the opposite side.
The bisector of a triangle is the bisector segment of an angle of a triangle connecting a vertex to a point on the opposite side.
The median of a triangle drawn from a given vertex is the segment connecting this vertex with the midpoint of the opposite side.
The midline of a triangle is the segment connecting the midpoints of its two sides.
Quadrilaterals.
A quadrilateral is a figure that consists of four points and four consecutive segments connecting them, and no three of these points should lie on the same line, and the segments connecting them should not intersect. These points are called the vertices of the triangle, and the segments connecting them are called its sides.
The sides of a quadrilateral starting from the same vertex are called opposite.
In a quadrilateral ABCD, vertices A and B are adjacent, and vertices A and C are opposite; sides AB and BC are adjacent, BC and AD are opposite; segments AC and WD are the diagonals of this quadrilateral.
Quadrilaterals can be convex or non-convex. Thus, the quadrilateral ABCD is convex, and the quadrilateral KRMT is non-convex.
Among convex quadrangles, parallelograms and trapezoids are distinguished.
A parallelogram is a quadrilateral whose opposite sides are parallel.
A trapezoid is a quadrilateral whose only two opposite sides are parallel. These parallel sides are called the bases of the trapezoid. The other two sides are called lateral. The segment connecting the midpoints of the sides is called the midline of the trapezoid.
BC and AD – bases of the trapezium; AB and CD – lateral sides; CM – midline of the trapezoid.
Of the many parallelograms, rectangles and rhombuses are distinguished.
A rectangle is a parallelogram whose angles are all right.
A rhombus is a parallelogram in which all sides are equal.
Squares are selected from many rectangles.
A square is a rectangle whose sides are all equal.
Circle.
A circle is a figure that consists of all points of the plane equidistant from a given point, which is called the center.
The distance from the points to its center is called the radius. A segment connecting two points on a circle is called a chord. The chord passing through the center is called the diameter. OA – radius, CD – chord, AB – diameter.
A central angle in a circle is a plane angle with a vertex at its center. The part of the circle located inside a plane angle is called the circular arc corresponding to this central angle.
According to new textbooks in new programs M.I. Moreau, M.A. Bantova, G.V. Beltyukova, S.I. Volkova, S.V. In the 4th grade, Stepanova is given construction problems that were not previously included in the elementary school mathematics curriculum. These are tasks such as:
Construct a perpendicular to a line;
Divide the segment in half;
Construct a triangle on three sides;
Construct a regular triangle, an isosceles triangle;
Construct a hexagon;
Construct a square using the properties of the diagonals of a square;
Construct a rectangle using the property of rectangle diagonals.
Let's consider the construction of geometric figures on a plane.
The branch of geometry that studies geometric constructions is called constructive geometry. The main concept of constructive geometry is the concept of “constructing a figure.” The main propositions are formed in the form of axioms and are reduced to the following.
1. Each given figure is constructed.
2. If two (or more) figures are constructed, then the union of these figures is also constructed.
3. If two figures are constructed, then it is possible to determine whether their intersection will be an empty set or not.
4. If the intersection of two constructed figures is not empty, then it is constructed.
5. If two figures are constructed, then it is possible to determine whether their difference is an empty set or not.
6. If the difference of two constructed figures is not an empty set, then it is constructed.
7. You can draw a point belonging to the constructed figure.
8. You can construct a point that does not belong to the constructed figure.
To construct geometric figures that have some of the specified properties, various drawing tools are used. The simplest of them are: a one-sided ruler (hereinafter simply a ruler), a double-sided ruler, a square, a compass, etc.
Different drawing tools allow you to perform different constructions. The properties of drawing tools used for geometric constructions are also expressed in the form of axioms.
Since the school geometry course deals with the construction of geometric figures using a compass and a ruler, we will also focus on the consideration of the basic constructions performed by these particular drawings with tools.
So, using a ruler you can perform the following geometric constructions.
1. construct a segment connecting two constructed points;
2. construct a straight line passing through two constructed points;
3. construct a ray emanating from the constructed point and passing through the constructed point.
The compass allows you to perform the following geometric constructions:
1. construct a circle if its center and a segment equal to the radius of the circle have been constructed;
2. construct any of two additional arcs of a circle if the center of the circle and the ends of these arcs are constructed.
Elementary construction tasks.
Construction problems are perhaps the most ancient mathematical problems; they help to better understand the properties of geometric shapes and contribute to the development of graphic skills.
The construction problem is considered solved if the method for constructing the figure is indicated and it is proven that as a result of performing the specified constructions, a figure with the required properties is actually obtained.
Let's look at some elementary construction problems.
1. Construct on a given straight line segment CD equal to a given segment AB.
The possibility of construction only follows from the axiom of delaying a segment. Using a compass and ruler, it is carried out as follows. Let a straight line a and a segment AB be given. We mark a point C on a straight line and construct a circle with a center at point C with a straight line and denote D. We obtain a segment CD equal to AB.
2. Through a given point, draw a line perpendicular to the given line.
Let points O and straight line a be given. There are two possible cases:
1. Point O lies on line a;
2. Point O does not lie on line a.
In the first case, we denote a point C that does not lie on line a. From point C as a center we draw a circle of arbitrary radius. Let A and B be its intersection points. From points A and B we describe a circle of the same radius. Let point O be the point of their intersection, different from C. Then the half-line CO is the bisector of the unfolded angle, as well as the perpendicular to the straight line a.
In the second case, from point O as from the center we draw a circle intersecting straight line a, and then from points A and B with the same radius we draw two more circles. Let O be the point of their intersection, lying in a half-plane different from the one in which the point O lies. The straight line OO/ is the perpendicular to the given straight line a. Let's prove it.
Let us denote by C the point of intersection of straight lines AB and OO/. Triangles AOB and AO/B are equal on three sides. Therefore, the angle OAC is equal to the angle O/AC, the two sides are equal and the angle between them. Hence the angles ASO and ASO/ are equal. And since the angles are adjacent, they are right angles. Thus, OS is perpendicular to line a.
3. Through a given point, draw a line parallel to the given one.
Let a line a and a point A outside this line be given. Let's take some point B on line a and connect it to point A. Through point A we draw a line C, forming with AB the same angle that AB forms with a given line a, but on the opposite side from AB. The constructed straight line will be parallel to straight line a, which follows from the equality of the crosswise angles formed at the intersection of straight lines a and with the secant AB.
4. Construct a tangent to the circle passing through a given point on it.
Given: 1) circle X (O, h)
2) point A x
Construct: tangent AB.
Construction.
2. circle X (A, h), where h is an arbitrary radius (axiom 1 of the compass)
3. points M and N of the intersection of the circle x 1 and straight line AO, that is (M, N) = x 1 AO (general axiom 4)
4. circle x (M, r 2), where r 2 is an arbitrary radius such that r 2 r 1 (axiom 1 of the compass)
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