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Paradigmatic View on the Concept of World Science  Volume 1

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August 21, 2020 Toronto, Canada 107 . SECTION V. PHYSICS AND MATHS DOI 10.36074/21.08.2020.v1.41 CONSTRUCTIVE METHOD OF SOLVING AND CREATING THE CONDITIONS OF MATHEMATICAL PROBLEM ABOUT UNKNOWN ANGLES IN A TRIANGLE Liudmyla Bezperstova senior teacher, teacher of physics and mathematics General Education School № 3 for Levels I-III specialized in Human Sciences named after V. O. Nizhnichenka Horishni Plavni, Poltava region Yurii Hulyi senior teacher, teacher of physics and mathematics General Education School № 2 Horishni Plavni, Poltava region Roman Bezperstov student of the 11th grade General Education School № 3 for Levels I-III specialized in Human Sciences named after V. O. Nizhnichenka Horishni Plavni, Poltava region UKRAINE Problems for finding unknown angles in a triangle are «inconvenient» to solve. The problems are not simple, standard calculations cannot be employed; they are quite difficult to solve in traditional, standard ways. Of course, there are ways to solve individual problems, which are often comprised of various auxiliary constructions that are difficult to think of, as well as consideration of several isosceles triangles. But if there are other angles in the problem, then the previous methods do not work. Here are some well-known problems. 1. Task № 337 from the geometry textbook by Atanasyan [1]. Point M is inside the isosceles triangle ABC with the basis BC, the angle MBC is 30°, the angle MCB is 10°. Find the angle AMC if the angle BAC is equal to 80° (fig. 1, a). The author gives instructions to make an additional construction: to find the point of intersection of the bisector of the angle A and the line BM. The solution of the problem includes several steps with additional constructions and consideration of several triangles and calculation of angles values. The conditions of such problems are presented in fig. 1 (b, c). 108 Paradigmatic view on the concept of world science Volume 1 . a b c Fig. 1. The problem from Atanasyan's textbook (a), a similar problem with other values of angles (b), a problem in which the point M is outside the triangle (c) 2. The problem of a triangle with angles values 20°, 80°, 80° («Bermuda» triangle in geometry) (fig. 2), [2]. Fig. 2. Two variants of the problem with unknown angle x and angles values 20°, 80°, 80° 3. The problem from the Ukrainian Mathematical Olympiad (fig. 3), [3]. The ABC triangle has the following properties: AB = BC, the angle ABC is equal to 100°. The points K and L are taken on the side АС, so that the angles КВА and LВС are equal 30°. The bisector of the angle CBA intersects the segment ВК at the point P. Find the value of the angle КРL. August 21, 2020 Toronto, Canada 109 . Fig. 3. The problem of finding an unknown angle in an isosceles triangle We offer a different approach to solving such problems. It is known that any triangle can be inscribed in a circumference. Then the given and unknown angles of the triangle can be central, inscribed, angles between chords, angles between intersecting circumferences. They are easy to identify if you use a regular polygon inscribed in a circumference. The values of the known angles of the triangle that occur in the problems are multiples of 10 (10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°). Therefore, it is convenient to use a regular octadecagon inscribed in a circumference (circumference is divided with the vertexes of the octadecagon into 360° the equal circular arcs ( = 20° ), then the inscribed angle resting on the side of 18 this polygon is 10°. Solving problems by finding unknown angles in a triangle using a regular polygon inscribed in a circumference contains a minimum number of steps. The elements of a triangle are considered as elements of a regular polygon. The sides and segments of a triangle are the diagonals of the polygon or their parts. All lines, containing segments of a triangle, pass only through certain points that are the vertexes of the polygon. This approach to solving problems is used by V.V. Prasolov [4] (problems 12.58 – 12.60). Application of the constructive method using an octadecagon to solve problem 1 [1]. Consider the ABC triangle as a triangle with vertexes 9-14-4. Diagonals 14-7, 9-15, 4-13, containing the segments BM, AM, CM, respectively, intersect at point M 4̆−9+13̆−15 100°+40° (fig. 4),. Unknown angle 퐴푀퐶 = = = 70°. 2 2 Fig. 4. The vertexes of the triangle ABC are the vertices of the octadecagon Problem 2 of the «Bermuda» triangle can be solved by placing an isosceles triangle so that the vertex opposite its basis coincides with the center of the circumference, and the other two – with points 1 and 18. (fig. 5). Then unknown 5̆−7+17̆−18 40°+20° angle is х = = = 30°. 2 2 110 Paradigmatic view on the concept of world science Volume 1 . Fig. 5. Placement of a triangle to a circumference: one vertex coincides with the center of the circumference, the other two - with the vertexes of the octadecagon The same method can be used to solve problem 3 [3] by placing the triangle 9̆−13+2̆−16 80°+80° as shown in fig. 6. Unknown angle is х = = = 80°. 2 2 Fig. 6. Two vertexes of a triangle lie behind a circumference, and the third coincides with vertex of the octadecagon, all sides of the triangle intersect the circumference at the vertexes of polygon. More important goal than solving the problem is finding the answer to the question, «How can you come up with such a problem?» According to psychologists, to solve a problem (task), you need to go beyond this problem (task). Everything in the world consists of certain components and is a part of something bigger, which often cannot be immediately imagined or realized as a separate element. The article provides an approach to so called «inconvenient» problems in the aggregate of their interrelationships with other constructions such as a circumference and a regular n-gon. The most important aspect for understanding and creating such problems is to find three diagonals, intersecting at one point, in the regular n-angle (for example, diagonals 4-13, 9-15, 7-14 in fig.4) [5]. Such diagonals are convenient to find in a regular octadecagon, triacontagon (its vertexes divide the circumference into equal arcs of 12°) or 36-gon (its vertexes divide the circumference into equal arcs of 10°), inscribed in a circumference. Using this approach, the authors created problems, examples of which are shown below in fig. 7: August 21, 2020 Toronto, Canada 111 . a b c Fig. 7. Problems with using triacontagon and 36-gon (a – isosceles triangle with angles 72°, 54°, 54°, b – isosceles triangle with angles 36°, 72°, 72°, c – triangle with point M, which is the point of intersection of the triangle bisector and the segment that divides the angle C into two angles 20° and 40° 112 Paradigmatic view on the concept of world science Volume 1 . The methods of solving problems 1, 2, 3, mentioned at the beginning of the article, are not applied for the problems shown in fig. 7. Problems with «uncomfortable» angles can be solved by using a minimum number of steps, using a constructive method with using an n-angle. The sequence of steps for solving problems by using such method are the following: • determine which n-angle is convenient to use for given angles; • choose the vertexes of the triangle so that at least two of them are the vertexes of the n-gon, and the drawn segments are parts of diagonals, intersecting at one point (fig. 8); • calculate the value of unknown angle, using the properties of the central angles, inscribed angles, and angles between chords and intersecting lines. Fig. 8. Demonstration of the application of a constructive method for solving the problem of finding an unknown angle using an octadecagon: different placement (a, b) of a triangle relative to circumference. Conclusion. The main point of the above considered problems is the discreteness of the angles values, which is determined by the number of sides of a regular n-gon. What is discreteness? The vertexes of an n-gon define at least two vertexes of a triangle. The remaining segments are the parts of the diagonals. If one of the vertexes does not lie on a circumference, then the two sides of the triangle contain the vertexes of the n-gon. The values of the angles are determined by the central inscribed angles and the angles between the chords. By using the constructive method with an n-gon, an unknown angle can always be found in a triangle, if some other angles are given. References: [1] Атанасян, Л.С., Бутузов, В.Ф. (2014). Геометрия. 7 – 9 классы: учеб. для общеобразоват. Организаций. 2-е изд. М.: Просвещение. [2] Математика. Задача про треугольник с углами 20, 80, 80. Бермудский треугольник в геометрии. Вилучено з https://www.youtube.com/watch?v=P-MifrOTIDk [3] Лейфура,В.М., Мітельман, І.М. (2003). Математичні олімпіади школярів України: 1991 – 2000 рр.: Навч.-метод. Посібник. К.: Техніка. [4] Прасолов, В. В. (1991). Задачи по планиметрии. М.: Наука. [5] Прасолов, В. В. (1991). Диагонали правильного 18-угольника. Квант,№ 5, С. 40 – 42. .
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