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Determine the equation of the perpendicular bisector of the segment AB, given A(1, -7) and B(6, -12).

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Answer to a math question Determine the equation of the perpendicular bisector of the segment AB, given A(1, -7) and B(6, -12).

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Esmeralda
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Para determinar a equação da reta mediatriz do segmento AB, que passa pelo ponto médio e é perpendicular ao segmento AB, precisamos dos seguintes passos:

Passo 1: Encontrar o ponto médio de AB.

O ponto médio M de AB é dado por:
M = \left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)

Substituindo os valores de A(1, -7) e B(6, -12):
M = \left(\dfrac{1 + 6}{2}, \dfrac{-7 + (-12)}{2}\right)
M = \left(\dfrac{7}{2}, \dfrac{-19}{2}\right)
M = \left(\dfrac{7}{2}, -\dfrac{19}{2}\right)

Portanto, o ponto médio M é M \left(\dfrac{7}{2}, -\dfrac{19}{2}\right) .

Passo 2: Determinar a inclinação da reta AB.

A inclinação da reta AB é dada por:
m_{AB} = \dfrac{y_2 - y_1}{x_2 - x_1}

Substituindo os valores de A(1, -7) e B(6, -12):
m_{AB} = \dfrac{-12 - (-7)}{6 - 1} = \dfrac{-5}{5} = -1

Portanto, a inclinação da reta AB é m_{AB} = -1 .

Passo 3: Determinar a inclinação da reta mediatriz.

A inclinação da reta mediatriz é o oposto da inclinação da reta AB e é perpendicular a ela. Então a inclinação da reta mediatriz é:
m_{mediatriz} = \dfrac{-1}{-m_{AB}} = \dfrac{-1}{-(-1)} = 1

Passo 4: Agora temos a inclinação da reta mediatriz e o ponto médio M. Podemos usar a equação ponto-inclinação para encontrar a equação da reta.

A equação ponto-inclinação é dada por:
y - y_1 = m(x - x_1)

Substituindo M \left(\dfrac{7}{2}, -\dfrac{19}{2}\right) e m_{mediatriz} = 1 na equação, obtemos:
y - \left(-\dfrac{19}{2}\right) = 1\left(x - \dfrac{7}{2}\right)
y + \dfrac{19}{2} = x - \dfrac{7}{2}
y = x - \dfrac{7}{2} - \dfrac{19}{2}
y = x - 13

Portanto, a equação da reta mediatriz do segmento AB é \boxed{y = x - 13} .

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