Demostração de Colinearidade e Equação de Reta.

por Floral Fury Ter 14 Dez 2021, 17:23

(IEZZI) Prove que os pontos

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são colineares e determine a equação de reta que os contém.

Resp.: Sem gabarito.

Boa tarde, colegas!
Fiz a questão da seguinte maneira: primeiro, demonstrei se eles estão sobre a mesma reta... e dps, fiz a equação de reta deles!
Gostaria de saber se o que eu fiz está certo, pois cheguei a procurar ela na internet, mas não encontrei nem o gabarito, e nem uma resolução sobre a mesma... Segue:
--------------------------
1º. (x₂ - x₁)(y₃ - y₂) = (x₃ - x₂)(y₂ - y₁) ----> (b-a)(b-c) = (c-b)(a-b) ------> b² - bc - ab + ac = ca - cb - ba + b² ----> 0 = 0

-------------------------
2º.

, dados que (a') é o coeficiente angular e (b') é o coeficiente linear.
Substituindo as coordenadas:

Subtraindo (II) de (I), chegamos que: (a') = -1
Logo, pegando o valor de (a') e jogando em (I)... temos que: (b') = a + b + c

Portanto, (r): -x + (a + b + c) = y
-----------------------------------------------------------------

Gostaria de saber se meus cálculos estão certos!
Obrigado! Very Happy

por **Renan Almeida** Qua 15 Dez 2021, 00:56

Acredito que segunda parte esteja correta, foi assim que eu resolvi e encontrei o mesmo resultado.

Já a primeira, não sei se entendi qual foi seu raciocínio. Você calculou considerando que os coeficientes angulares sejam iguais?

Eu fiz da seguinte forma:
Montei uma matriz com as coordenadas dos pontos(a mesma que montamos para calcular a área de um triângulo dados seus vértices).
Caso os 3 pontos sejam colineares, o determinante dessa matriz deve ser igual a 0, pois a área do "triângulo" será nula, afinal é apenas uma linha.

[latex]D = \begin{vmatrix} a & b+c & 1\\ b &a+c & 1\\ c &a+b &1 \end{vmatrix}[/latex]

[latex]D = a(a+c) + c(b+c) + b(a+b) - [c(a+c)+b(b+c)+a(a+b)][/latex]

[latex]D = a^2 + ac + c^2 + bc + b^2 + ab - ac - c^2 - bc - b^2 - ab - a^2[/latex]

[latex]D = 0[/latex]

por Floral Fury Qua 15 Dez 2021, 10:02

Olá colega Renan!

Sobre a segunda parte, fico feliz pq eu n achei nenhum gabarito... então pelo menos mais alguém achou o mesmo resultado que eu kkkkk.

Sobre a primeira parte: Essa expressão que eu usei é dada pelo Iezzi para quando queremos saber se os pontos estão sobre a mesma reta! Nessa expressão, ele n considera nada sobre coeficientes angulares/lineares não... Ele apenas usa das coordenadas como um método para se discutir sobre a colinearidade desses pontos dados!

No meu caso, deu a identidade 0 = 0... Logo, podemos provar q eles estão na mesma reta!
E no seu caso, vc fez por determinante, e chegou no mesmo resultado!

Obrigado! Very Happy