Prova  de  Matemática 3º Ano:     Geometria  Analítica.                   Professor:  Irenilson  Santos        Ano letivo: 2012
Autor: ESCOLA ESTADUAL OSWALDO CRUZ | Publicado: 2012-03-14T14:08:00-07:00







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LEIA&nbsp; ATENTAMENTE&nbsp; AS&nbsp;
INSTRUÇÕES&nbsp; SEGUINTE:



1.Esta prova contém 10 questões de geometria analítica.

2.Escreva todo tipo de solução na folha de &nbsp;resposta.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ATENÇÂO:
após a conferência, escreva seu nome na folha de resposta



QUESTÕES:

&nbsp;

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<![endif]--><!--[if gte mso 10]>

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 mso-style-priority:99;
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 mso-style-parent:"";
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 mso-para-margin-top:0cm;
 mso-para-margin-right:0cm;
 mso-para-margin-bottom:10.0pt;
 mso-para-margin-left:0cm;
 line-height:115%;
 mso-pagination:widow-orphan;
 font-size:11.0pt;
 font-family:"Calibri","sans-serif";
 mso-ascii-font-family:Calibri;
 mso-ascii-theme-font:minor-latin;
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 mso-hansi-theme-font:minor-latin;
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<![endif]-->



01.Consideremos os pontos, A e B, do sistema cartesiano
ortogonal indicado na figura.&nbsp;&nbsp; Demonstre
a fórmula matemática que relaciona a Distância entre os pontos A e B:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp;<!--[if gte mso 9]>
 
  12.00
 
 
  
  
 
<![endif]--><!--[if gte mso 9]>
 
  Normal
  0
  
  
  21
  
  
  false
  false
  false
  
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<![endif]--><!--[if gte mso 9]>
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02.Um grupo de jovens acampou às margens de um rio, montando
sua barraca no ponto médio do segmento que une as árvores A( -1, 4) e B(5, 2).
Determine as coordenadas do ponto onde eles acamparam.

 



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03. &nbsp;Admita que um arquiteto construa a maquete de
uma praça no formato de um triângulo escaleno e decide colocar um monumento num
determinado ponto, no interior da praça. Ele marca, inicialmente, o ponto médio
de cada lado da praça e, a partir de cada ponto obtido, traça uma linha reta
até o vértice oposto ao ponto médio marcado colocando na intersecção dessas
linhas retas o monumento. A partir dessas informações, é correto afirmar que o
ponto onde o monumento será colocado denomina-se:

 A)
Incentro do triângulo&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; B)
Ortocentro do triângulo&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
C) Baricentro do triângulo&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp; D)
Circuncentro do triângulo&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; E) Altura
do triângulo





04. &nbsp;&nbsp;Na arquitetura, a Matemática é usada em todo
momento. A Geometria é especialmente necessária no desenho de projetos. Essa
parte da Matemática ajuda a definir a forma dos espaços, usando as propriedades
de figuras planas e sólidas. Ajuda também a definir as medidas desses espaços.
Uma arquiteta é contratada para fazer o jardim de uma residência, que deve ter
formato triangular. Analisando a planta baixa, verifica-se&nbsp; que os vértices possuem coordenadas A(8, 4),
B(4, 6) e C(2, 4). No ponto médio do lado formado pelos pontos A e C, é
colocado um suporte para luminárias. Considerando o texto e seus conhecimentos,
determine:









A) A distância do suporte ao ponto B;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; B) A área do 
 
 
  
  
  
  
  
  
  
  
  
  
  
  
 
 
 

 
ABC



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<![endif]--><!--[if gte mso 9]>
 
  
 <![endif]-->



05. 
 
 
  
  
  
  
  
  
  
  
  
  
  
  
 
 
 

 
 
&nbsp;&nbsp;Quatro cidades, A, B, C e D, foram construídas
à beira de uma rodovia reta conforme a ilustração abaixo: A distância entre A e
C é de 50 km e a distância entre B e D, é de 45 km. Além disso, sabe-se que a
distância entre a primeira e a última é de 80 km. Então temos:

&nbsp; 

A) AB = 15km, BC = 35km e CD = 30km &nbsp; B) AB = 35km, BC = 15km e
CD = 30km&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; C) AB = 30km, BC = 15km e CD = 35km&nbsp;&nbsp;&nbsp;&nbsp; D) AB = 35km, BC = 30km e
CD = 15km



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06. &nbsp;A figura
geométrica cujo contorno é definido pelos pontos (1,1), (3,2), (5,1) e (3,5) do
plano cartesiano tem sua forma semelhante a&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; A)
uma ponta de flecha.&nbsp;&nbsp;&nbsp; &nbsp; B)
uma bandeirinha de festa junina.&nbsp;&nbsp; C) uma tela de televisão.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; D) uma prancha de surfe.&nbsp;&nbsp;&nbsp; E) um polígono regular.



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07. Um sistema cartesiano é associado à planta de
uma cidade de modo que o eixo Ox é&nbsp;
orientado de Leste para Oeste, e o eixo Oy é orientado de Sul para Norte
e a unidade adotada em cada eixo é o quilômetro. Um automóvel que parte de um
ponto A do terceiro quadrante distante 3 km do eixo Ox e 5 km do eixo Oy&nbsp; percorre o seguinte trajeto: 15 km para o
leste; 3 km para o&nbsp; norte; 3 km para o
oeste e finalmente, 2 km para o norte, estacionando em um ponto B.&nbsp; Em relação e esse sistema de coordenadas, determine
a distância, em km, entre os pontos A e B &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;08.
Os
pontos A e B da figura são vértices do triângulo ABC, retângulo em A. O vértice
C pertence ao eixo das ordenadas. Determine as coordenadas do ponto C. 



















&nbsp;



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09. Classifique o 
 
 
  
  
  
  
  
  
  
  
  
  
  
  
 
 
 

 
ABC quanto aos lados e quanto aos
ângulos, sabendo que A(1,-2), B(-3,1) e C(1,4).



10. O vôo 001 de uma companhia aérea parte da cidade A, faz uma
escala na cidade B e, de lá, segue para seu destino final, que é a cidade C.
Quando representamos essas cidades em um plano cartesiano com origem na cidade
B, tendo o quilômetro como unidade de medida e usando uma escala de 1:40.
Quantos quilômetro essa aeronave percorre nesse trajeto sabendo que A(-12,5)
e&nbsp; C(9,12)?