Geometria Hiperbólica por meio do soffware NonEuclid
Autor: Matematica | Publicado: 2011-11-14T19:12:00-08:00

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 <![endif]-->  &nbsp;A atividade a seguir possibilita maior interação com a geometria hiperbólica, visto que com o software é possível visualizar com maior precisão o comportamento de retas, segmentos de retas no plano e também na composição de figuras geométricas. <!--[if gte mso 9]>     Normal   0            21         false   false   false      PT-BR   X-NONE   X-NONE                                                                                                     <![endif]--><!--[if gte mso 9]>  <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
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 <![endif]-->A idéia de se utilizar esse software no ensino de geometria hiperbólica é de deixar os alunos livres para manipulá-lo e descobrir certas proposições e teoremas através de questões levantadas pelos professores.   
ATIVIDADESDefinição: Um triângulo isósceles é um triângulo que possui pelo menos dois lados de mesmo comprimento. O terceiro lado do triângulo isósceles é chamado de base.
Construção de um Triângulo Isósceles:1.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Dados três pontos não colineares A, B e C (Comando “Traçar ponto” menu “Construções”). 2.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Construa um circulo com centro em A. (Comando “Traçar círculo” menu “Construções”).3.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Construa a semi reta AB (com extremidade em A que passa por B). (Comando “Traçar semi-reta” menu “Construções”).4.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Construa a semi reta AC (com extremidade em A que passa por C ).5.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Seja E a interseção da semireta  AB e o circulo de centro em A. (Comando “Traçar ponto de Interseção” menu “Construções”).6.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Seja H a interseção da semireta AC  e o circulo de centro em A .7.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Construa o triângulo AHE . Por construção, o triângulo AHE&nbsp;é isóceles visto que dois lados são raios do mesmo círculo. (Comando “Traçar segmento de reta” menu “Construções”).
Atividade 1 – Triângulo Isósceles    1.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Encontre a medida dos ângulos da base deste triângulo (Comando “Medir ângulo” menu “Medidas”). O que se pode observar? R: Que eles são iguais.2.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; A altura de um triângulo isósceles divide a base em segmentos congruentes?&nbsp; (Comando “Traçar bissetriz” menu “Construções”, selecione os pontos que constituem os vértices. Encontre o ponto de interseção da bissetriz com a base, comando “Traçar interseção” menu “Construções”)R: Sim.3.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; A partir deste triângulos use o comando “Mover ponto” do menu “Editar” para construir um triângulo eqüilátero e responda:Se um triângulo é eqüilátero então é eqüiângulo? Se um triângulo é eqüiângulo, então é eqüilátero? A medida de cada ângulo interno de um triângulo eqüilátero é de 60º?R: Um triângulo equilátero é equiângulo e vice-versa, porém seus ângulos são menores que 60º.
Atividade 2 – Triângulo Retângulo1.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Construa um triângulo retângulo. Primeiramente trace 2 pontos A e B.&nbsp; (Comando: “Traçar ponto” menu “Construções”). Em seguida ligue os pontos A e B. (Comando: “Construções” – “Traçar segmento de reta”). Trace uma reta perpendicular a AB em B. (Comando: “Construções” – “Traçar perpendicular”. Ligue o segmento AC.2.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; O teorema de Pitágoras se verifica nesta geometria?R: Não. Nesse caso o quadrado da hipotenusa é maior que a soma dos quadrados dos catetos.

Atividade 3 – Quadriláteros
Construção dos quadriláteros da geometria hiperbólica:
Quadrilátero de Saccheri:1.&nbsp; Primeiramente criamos dois ponto A e B (menu “Construções” no comando “marcar ponto”), ai por esses dois pontos passamos uma reta ( menu “Construções” e comando “traçar retas”);2.&nbsp; Criamos duas retas perpendiculares uma passando pelo ponto A e outra pelo ponto B (menu “Construções” e comando “traçar perpendicular”);3.&nbsp; Criamos uma reta que passe pelos pontos C e D (menu “Construções” e comando “traçar retas”);4.&nbsp; Podemos medir os ângulos do quadrilátero formado (menu “Medidas” e comando “medir ângulos”).
Quadrilátero de Lambert:1.&nbsp; Primeiramente criamos dois ponto A e B (menu “Construções” e comando “marcar ponto”), e por esses dois pontos passamos uma reta ( menu “Construções” e comando “traçar retas”);2.&nbsp; Criamos duas retas perpendiculares uma passando pelo ponto A e outra pelo ponto B (menu “Construções” e comando “traçar perpendicular”);3. Criamos outra perpendicular passando pelo ponto C (menu “Construções” e comando “traçar perpendicular”), sendo que para traçar esta reta perpendicular clicamos no ponto C e na reta BD.4.&nbsp; Podemos medir os ângulos do quadrilátero formado (menu “Medidas” e comando “medir ângulos”).