Page 344 - PowerPoint Presentation

KONSTRUKCIJE UGLOVA I MNOGOUGLOVA SA UNIVERZALNIM OPISOM
CONSTRUCTION OF ANGLES AND POLYGONS WITH UNIVERSAL DESCRIPTION

Autor: Andrea Popović 7. razred OŠ"Jelena Ćetković"
Mentor: Violeta Komnenović, diplomirani matematiĉar ETŠ "Nikola Tesla", Beograd

Rezime
Rad sadrţi konstrukcije uglova, trouglova po stavovima podudarnosti, paralelograma, trapeza i pravilnih
mnogouglogova.Konstrukcije uglova 30˚, 45˚, 60˚i 90˚ predstavljaju osnovu za konstrukcije svih drugih
uglova (75˚,135˚,120˚…). Stavovi podudarnosti trouglova su SSS (stranica, stranica,stancica), gde su
poznate sve tri duţinestranica trougla, SUS(stranica, ugao, stanica), gde su nam poznate duţine dve
stranice i ugao koje zaklapaju, SSU (stanica, stanica, ugao), gde su nam poznate duţine dve stranice i
ugao na jednoj od njih i USU(ugao, stanica, ugao) gde nam je poznata stranica i uglovi na njoj. Za
konstrukciju paralelograma potrebna su tri nezavisna elementa. Kvadrat je odreĊen jednim nezavisnim
elementom (poznato je da su mu svi uglovi pravi 90˚, i dijagonale mu se seku pod pravim uglom). Za
konstrukciju trapeza potrebna su nam ĉetiri nezavisna elementa, a za konstrukciju pravouglog trapeza tri.
Za konstrukcije pravilnih šestouglova, osmouglova i dvanaestouglova dovoljan je jedan elemenati znanje
formula za dobijanje neophodnih uglova. Konstrukcije su kosturom predstavljene analizom koja sadrţi i
skicu, samom konstrukcijom i detaljnim opisom konstrukcije. Stavovi podudarnosti sadrţe dokaz kojim
potvrĊujemo da je konstruisani trougao, traţeni, kao i diskusiju o dva podudarna ili jedinstvenom
rešenju. Rad pokazuje univerzalnost primene matematiĉkih konstrukcija, oznaka i simbola. TakoĊe da za
konstrukcije nisu dovoljni samo šestar i lenjir vec mnogo šire znanje iz razliĉitih oblasti na jednom mestu.
Skice i konstrukcije su raĊene u “GeoGebra” dok je deo konstrukcija ruĉno raĊen. Ostvareni rezultati
ispitivanja ove oblasti su sledeći: pristupacan nacin komunikacije meĊu matematiĉarima,deo geometrije
vrlo zanimljiv za obradu, “most” koji spaja svu decu sveta.
Kljuĉne reci: Analiza, skica, konstrukcija, opis konstrukcije, lenjir , šestar.

Summary
This work contains constructions of angles and triangles according to the positions of congruence,
parallelogram, trapezoid andregular polygons.Constructions of angles 30 ˚,45 ˚,60˚ and 90 represent the
base for constructions of all the other angles (75 ˚,135 ˚,120…). Positions of congruence of triangles are
SSS (side, side, side) where all three of triangle sides are familiar, SAS (side, angle, side) where we are
familiar with two sides and an angle on one of them, and the last but not the least ASA (angle, side,
angle) where we are familiar with one side and its angles.For the construction of parallelogram we need
three independent elements. Square is defined with one independent element (we are familiar with the
fact that all of its angles are right 90˚, and its diagonals are cut under the right angle).For the construction
of trapezoid we need four independent elements, and for the construction of right-angled triangle we need
three independent elements. Only one element and the knowledge of formulae for getting the necessary
angles is enough for the construction of right-angled hexagons, octagons and others. These framework
constructions are presented with the analysis which contain sketch, construction itself and detailed
description of construction. Positions of congruence have the proof with which we confirm that the
constructed triangle is sought, as well as the discussion of two congruent angles or unique solution. This
work also shows the universality of the use of mathematical constructions, marks and symbols. In on
place, it also shows that you do not only need a pair of compasses and ruler but much wider knowledge of
different areas. Sketches and constructions are done in “GeoGebra ”while the part of the constructions are
hand written. Achieved results, which were the issue of this search, are: accessible way of communication
among mathematicians, segment of geometry which is interesting for working on, and “the bridge” which
connects the children from all over the world.
Key words: Analysis, sketch, construction, description of construction, a ruler, a pair of compasses.