STUDY OF THE METHOD OF SIMILARITY IN SPACE
Anna Gennadievna Zvyagintseva
3rd year student of the faculty of Mathematics and Natural Science Education
Belgorod State National Research University
Belgorod
Scientific supervisor
Alexander V. Markov
Senior lecturer
Foreign languages department
NRU BSU
Annotation
This article is devoted to the study of the method of similarity in space when solving problems. The main axioms of stereometry, the practical application of the similarity method in solving spatial problems are considered. The concepts of lines, planes and parallelism are analyzed.
Key words: method of similarity in space, problem solving, stereometry, study of the method of similarity.
Subbiosis theory is a method of mathematical modeling based on the transition from ordinary physical quantities affecting the system being modeled to generalized quantities of complex type composed of the original physical quantities, but in certain combinations depending on the specific nature of the process under study.
In the course of his life and work, man, in one way or another, investigates the shape, size, and location of objects in space without ceasing. Similar problems are solved by astronomers who work with scale objects, as well as by physicists who study the structure of particles at the molecular level. The section of geometry that studies such problems is called stereometry.
It can also be said that the plane inherently does not have any unambiguous meaning in geometry. So, if, for example, there is a point in the plane of parallelepiped, then it is logical to assume, that there are similar points beyond the boundaries of the plane of this figure [10, p.33].
In planimetry this thesis disappears by itself, because all objects and points are located in one plane, which is the only one.
Stereometry is another case: there are several planes. In each of them the same postulates as in planimetry take place, but the properties of the planes themselves need a separate analysis.
It is not uncommon to encounter problems where the condition breaks down into parts. This is where the similarity method is very effective. The first part of the condition identifies the sought figure with exact similarity. The other part of the condition makes it possible to choose from analogous figures the one that corresponds to it, and thus represents the solution of the problem.
In stereometry one more axiom is added to the key theses of planimetry - the plane, and also the axioms, which study the arrangement of planes in relation to other figures of geometry [8, p.15]. Three such axioms have been established.
The first one, the axiom of space emergence, creates a third dimension in space. It sounds as follows:
- There are 4 points that do not lie in one plane (Fig. 1).
Figure 1 - Four points not in the same plane
It turns out that not all points are in the same plane. But this is not enough. It is necessary that there should be an infinite number of planes different from each other. This is reinforced by the following axiom, the axiom of the plane. It sounds like this:
- A plane passes through every 3 points.
The third axiom takes place when folding of paper figures takes place. Clearly the fold lines are straight lines.
The axiom of the intersection of planes is as follows:
When two planes have a common point, their intersection is a straight line (Fig. 2).
Figure 2 - Graphical representation of the axiom of intersection of planes
In view of this we can say a corollary: if three points are on the same line, then the plane passing through them is the only one.
And it is true that if two different planes pass through some 3 points, then it is possible to draw a line through these points on which the planes intersect. This property is often included in axioms.
The third axiom is extremely important for the understanding of stereometry: it endows the space with three-dimensional directionality, because in spaces of dimension four and above the planes are capable of intersecting by one point [6, p.24]. To the three considered planimetric axioms are added, corrected taking into account the fact that now there is not one, but several planes. Thus, the line axiom - it is possible to draw a single line through two different points - is fixed in stereometry in advance, but only it already extends to 2 points of space.
As a consequence, a useful property follows directly from the axioms: a line which has at least 2 points in common with the plane is completely located in the given plane.
Suppose that the line l passes through points A and B of the plane α (Fig. 3). Outside this plane there exists at least one point C (by the axiom of space-exit). According to the axiom of the plane, the plane β can be drawn through A, B, and C. It is different from the plane α because it includes C and has two points in common with α. Accordingly, β intersects α by a line which, like l, belongs to A, B. By the axiom of the straight line, the line of intersection of the planes coincides with l. But this line lies in the plane α, which was required to prove.
Figure 3 - Location of line l in the plane α
As a result of the study, we can say that all the statements of planimetry are realized on any plane.
The similarity method for solving spatial problems consists in the fact, that firstly, using only part of the data, construct a figure similar to the sought figure, and then, using the remaining data, construct the sought figure [3, p.15]. Let's consider the first example of the solution of the construction problem by the similarity method.
Problem. Construct triangle ABC with defined acute angle B, where AB:BC = 3:2 and height CD is equal to the given segment PQ.
Solution. On the sides of the set angle B lay the segments BA1 and BC1, equal respectively to 3PQ and 2PQ (Fig. 11, a). Triangle A1BC1 is similar to the sought triangle by the first sign of similarity of triangles. If its height C1D1 is equal to PQ, then A1BC1 is the required triangle.
Figure 11 - Graphical interpretation of the similarity method when solving spatial problems
Suppose that C1D1 ≠ PQ. The sought point C is located from the line BA1 at a distance equal to PQ, i.e., it belongs to the set of points located from the line BA1 at a distance PQ. Correspondingly, C belongs to the line parallel to BA1 at distance PQ. Construct this line (line a in Fig. 11b) and mark by C its intersection point with the line BC1.
Through C draw a line parallel to A1C1 passing through BA1 at a certain point A. Triangle ABC is the desired triangle.
Actually, angle B is original, altitude CD is PQ, and since AC || A1C1, triangles ABC and A1BC1 are congruent, therefore AB:A1B = BC:BC1, and consequently AB:BC = A1B:BC1 = 3:2.
The similarity method is used mainly in those situations, when among the given conditions only one represents a segment, and all the others are either angles or relations of segments.
Most often it is more effective to construct an auxiliary figure so that it is similar not only to the searched figure, but also similarly situated with it. The quality of problem solution in such a situation depends on the choice of similarity center.
When solving construction problems by the similarity method, it is not uncommon to apply the following observation. In the case when two figures are similar, the similarity coefficient is equal to the ratio of any pair of set segments [2, p.18]. If segments a, b, c,... of the figure F correspond to the segments a1, b1, c1,... of the similar figure F1, then the similarity coefficient is also equal to the relations:
Problem. Given ABC and point M is situated inside it. Find point X on the side BC being at equal distances from line AB and point M.
Solution. Let point X be found such that the perpendicular XY = MX. The problem is reduced to the construction of the figure ҮХМ. We establish a list of figures similar to the sought figure. One can only construct one of these figures, e.g. RKN, since it remains to draw a line parallel to CR from point M and the problem will be realized.
For the construction of figure RKN it is necessary to say that B is the center of similarity of figures and because of this the points M, H, K and B are located on one straight line BM and PNAB, PN = BN, the finding of point P takes a random character. Therefore to construct PKN it is necessary to restore PNAB at point P and to describe an arc from center N with radius PN which intersects BM in point K. Passing MH║KN we can find the required point X.
Stages of construction:
ЕGAB;
H = ω (G, EG)BM;
MX║HG;
X = BCMX.
Proof. By drawing the perpendicular XY we obtain MH: GH = BX: BN = XY: GE, whence MH: GH =XY: GE, but since by construction HG = GE, then MH = YX.
Investigation. The problem is always possible and has two solutions because the arc from center G meets B always in two points.
Consider one more problem.
Task. It is necessary to construct triangle ABC if the ratio AB:BC, ABC and radius of the incircle are known.
Solution. Since the angle and the ratio of sides of the angle in the required triangle are set, leave the other conditions alone and construct a triangle similar to the required one. For this purpose, on the sides of the angle set, lay BD equal to m equal parts and BE equal to n equal parts, and connect points D and E. Then the desired triangle and triangle DBE are similar because they have an equal angle between the proportional sides. Plotting the segments parallel to DE in angle ABC we obtain triangles similar to the required one, but with different radii of incircle; of all these triangles it is necessary to choose one with radius of incircle equal to r. Having established the center O, it is easy to construct the triangle itself.
Steps of construction:
1. OFDE;
OG = r;
Through G draw AC║DE;
∆ABC is the desired.
Proof. It follows from the construction.
Research. There is always one possible solution.
So, stereometry is a section of geometry that studies the properties and arrangement of objects in space. By studying a variety of geometric figures - imaginary objects, you can get an idea of geometric features of the studied real figures (their shape, relation to each other, etc.) and apply the obtained knowledge in practice. Stereometry is constantly used in construction, astronomy, architecture, geography and other sciences and spheres of human activity.
In our article we looked at the method of similarity in solving problems in space. It is also possible to understand that in stereometry, a competently made drawing is able to be not just a picture, but the foundation of problem solving.
List of references
1. Alexandrov A.D.: The beginnings of stereometry, 9. - Moscow: Prosveshcheniye, 1981.
2. L.S. Atanasian: A Course in Elementary Geometry. - M.: Santax-Press, 1997.
3. Beskin, L.N.: Stereometry. - Moscow: Prosveshcheniye, 1971.
4.Bogomolov N.V.: Mathematics. - Moscow: Drofa, 2002.
3.Vasilevsky A.B.: Method of Similarity. - Minsk: Narodnaya Asveta, 1985.
4.Pogorelov A.V.: Elementary Geometry. - Moscow: Nauka, 1977.
5.Prasolov V.V.: Tasks in stereometry. - Moscow: Nauka, 1989
6. Solodovnik L.F.: Stereometric Tasks (using Shamanov V.P. Device). - Belgorod: Belgorod State University, 2002.
7. Comp. A.A. Egorov: Such different geometry. - Moscow: Quantum Bureau, 2001.
8.Khakhamov L.R.: Transformations of the plane. - Moscow: Prosveshcheniye, 1979.
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