PLANE DILATIONS IN THE SCHOOL MATHEMATICS COURSE AND THE THEORY OF GEOMETRIC TRANSFORMATIONS FOR FUTURE MATHEMATICS TEACHERS
DOI:
https://doi.org/10.31110/fmo2026.v41i2-07Keywords:
geometric transformations, similarity transformations, dilation, homothety, analytical geometry, school geometry curriculum, extracurricular activities in schools, future mathematics teacherAbstract
Formulation of the problem. Geometric transformations constitute an important component of the school geometry curriculum and the professional training of future mathematics teachers. However, an analysis of school textbooks and educational literature reveals certain methodological inaccuracies in the interpretation of geometric transformations and their properties. This necessitates a more systematic, scientifically grounded presentation of similarity transformations and their particular subgroups in analytical geometry courses for pre-service mathematics teachers.
Materials and methods. The study employs theoretical methods of scientific and pedagogical research, including the analysis and synthesis of scientific and educational literature, as well as mathematical methods for proving and substantiating geometric statements.
Results. The article examines plane dilations as a subgroup of similarity transformations. Their fundamental properties and invariants — geometric properties of figures and relations between them that are preserved under the corresponding transformations — are analyzed. The connection between dilations, homotheties, and translations of the plane is established. It is shown that dilations form a group of geometric transformations. In addition, the paper presents theoretical statements and examples of problems illustrating the application of these transformations in the study of geometric objects.
Conclusions. The conducted research enables the systematization of the concept of plane dilations as a subgroup of similarity transformations and clarifies their place within the structure of geometric transformations. The results can be used in teaching analytical geometry at higher education institutions and in school geometry courses, particularly in elective classes, mathematical circles, and extracurricular activities. Prospects for further research include expanding the analysis of subgroups of similarity transformations and their application in the methodology of teaching geometry.
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