Respondendo alguns por quês matemáticos
Material preparado para a oficina no Workshop 2020
Table of Contents
- Áreas
- Estimativa da medida da área do território do Brasil
- Área do Retângulo
- Qual é a fórmula da Área do Paralelogramo? Por que?
- Qual é a fórmula da Área do triângulo? Por quê?
- Qual a fórmula da Área do trapézio? Por quê?
- Qual a fórmula da Área do Losango? Por quê?
- Círculo
- Área do Círculo
- Pontos Notáveis do Triângulo
- Por que o Incentro equidista dos lados do triângulo?
- Por que o Circuncentro equidista dos três vértices do triângulo?
- Geometria Espacial
- Dedução da fórmula do Volume da esfera
Estimativa da medida da área do território do Brasil
A área terrestre do Brasil
O Brasil, levando em consideração seu território, é o 5º maior país do mundo. Mas o que isso significa? Qual é o tamanho do Brasil?
Tentando fazer uma estimativa da área
Observe os dois mapas seguintes
[url=http://geoconceicao.blogspot.com/2011/02/pontos-extremoscoordenadas-geograficas.html]http://geoconceicao.blogspot.com/2011/02/pontos-extremoscoordenadas-geograficas.html[/url]
Reflexão
Analisando os mapas anteriores, daria para fazer uma estimativa de qual é o tamanho do território brasileiro em Km²? Qual a medida que você sugeriria? Pode usar: "está entre tanto e tanto"; "pelo menos"
Melhore a estimativa usando a malha quadriculada
No applet seguinte, movimente os seletores B e H de forma que os pontos pretos sejam vértices de uma região retangular que compreenda todo o território do Brasil. Depois altere o seletor n de forma que preencha todo o território brasileiro. Considerando que cada quadradinho tenha lado medindo 1000 km, daria para fazer uma estimativa melhor sobre o tamanho do território brasileiro em Km²? Qual a medida que você sugeriria? Pode usar: "está entre tanto e tanto"; "pelo menos"
Área do Círculo
Por que o Incentro equidista dos lados do triângulo?
Incentro
O incentro de um triângulo é o ponto de encontro das bissetrizes dos ângulos internos do triângulo.
Construção do Incentro do Triângulo
[list][*]Ative a ferramenta POLÍGONO (Janela 5) e clique em três lugares distintos para formar um triângulo. Para fechar o triângulo clique novamente no primeiro ponto. Naturalmente que os pontos não podem estar alinhados. Um triângulo com vértices nos pontos A, B e C será criado.[/*][*]Ative a ferramenta BISSETRIZ (Janela 4) e clique sobre os vértices: A, C e B (nessa ordem). Posteriormente sobre os vértices C, B e A (nessa ordem). Duas bissetrizes foram criadas com os nomes “d” e “e”.[/*][*]Ative a ferramenta INTERSEÇÃO DE DOIS OBJETOS (Janela 2) e crie o ponto D de interseção das retas “d” e “e”. [br][/*][*]Queremos traçar a terceira bissetriz. A pergunta é: será que ela também passará pelo ponto D? Ative a ferramenta BISSETRIZ (Janela 4) e clique nos pontos B, A e C (nessa ordem).[br][/*][/list]
Construção do círculo inscrito
[list][*]Ative a ferramenta EXIBIR/ESCONDER OBJETO (Janela 12), clique sobre as retas d, e, f e aperte ESC posteriormente. [br][/*][*]Vamos modificar o nome do ponto D para Incentro. Para tal, clique com o botão do lado direito do mouse sobre o ponto D e selecione a opção RENOMEAR. Na nova janela que aparecerá, escreva Incentro e clique em OK.[/*][*]Ative a ferramenta RETA PERPENDICULAR (Janela 4), clique no ponto Incentro e no lado c do triângulo (que liga os pontos A e B). [br][/*][*]Ative a ferramenta INTERSEÇÃO DE DOIS OBJETOS (Janela 2), clique na reta g e, posteriormente, no lado c que liga os pontos A e B. Um ponto D será criado[1]. [br][/*][*]Nosso interesse é apenas no pé da perpendicular (ponto D). Assim, podemos esconder a reta g, usando a ferramenta EXIBIR/ESCONDER OBJETO (Janela 12). Feito isto, aperte a tecla ESC. [br][/*][*]Ative a ferramenta CÍRCULO DEFINIDO PELO CENTRO E UM DE SEUS PONTOS (Janela 6), clique no ponto Incentro e posteriormente no ponto D. Uma circunferência h será criada.[/*][/list]
Reflexão 1
Por que os lados do triângulo tangenciam o círculo?
Reflexão 2
Meça as distâncias de um dos vértices do triângulo a dois pontos de tangência e observe que são iguais. Por que?[br][br] [img 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[/img]
Propriedade
As três bissetrizes internas de um triângulo interceptam-se num mesmo ponto que equidista dos 3 lados do triângulo.
Demonstração
Dedução da fórmula do Volume da esfera
Movimente o controle deslizante "Etapas" e marque as caixas de Esconder/Mostrar
Reflexão
Clique com o botão direito do mouse, segure e arraste a janela 3d. Olhe a anticlepsidra em diferentes posições. Ela se parece com algum objeto que você conhece?
Áreas das secções
Reflexão 2
Movimente o ponto R para alterar a posição do plano. Compare as medidas das áreas da coroa circular e do círculo. O que você percebe?
Justificativa?
Para justificarmos a propriedade anterior, precisamos verificar outra.
Raio do círculo menor da coroa e distância do centro ao vértice do Cone
Reflexão 3
Na construção anterior, compare a medida do raio do círculo menor da coroa com a distância de seu centro ao vértice. O que você observa? Altere a posição do ponto R para movimentar o plano. O que você observa?
Exercício-Justificativa
Utilize a figura seguinte para justificar a propriedade vista anteriormente. Use semelhança de triângulos para mostrar que o raio do círculo menor da coroa é igual a distância de seu centro ao vértice. Lembre-se que a altura do cone é igual ao raio da base. [br][br][img]https://www.geogebra.org/resource/y3tMTyC4/BgBntHKRJW7SAskQ/material-y3tMTyC4.png[/img]
Justificativa para as secções com mesma área
Reflexão 1
Entendeu a demonstração? Desmarque as caixas "Exibir/Esconder" para visualizar melhor as atividades.
Volume da anticlepsidra
Mostramos que as medidas das áreas das secções formadas na anticlepsidra e na esfera são iguais. Como a altura da anticlepsidra é igual ao diâmetro da esfera, então pelo princípio de Cavalieri, possuem volumes iguais. Dessa forma, basta calcular o volume da anticlepsidra. Para isso, precisamos calcular o volume do cilindro e subtrair o volume dos dois cones.
Conclusão
Dessa forma, o volume da esfera é [br][b][center][math]\resizebox{6cm}{1cm}{V=\frac{4\pi R^3}{3}}[/math][code][/code][/center][/b]