To determine the slope of a line that is perpendicular to the line given by the equation [tex]\((y - 3) = 5(x - 4)\)[/tex], we begin by identifying the slope of the given line [tex]\(AB\)[/tex].
The equation [tex]\((y - 3) = 5(x - 4)\)[/tex] is already in the point-slope form of a linear equation, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line.
From this equation, we can see that the slope [tex]\( m \)[/tex] of the line [tex]\(AB\)[/tex] is [tex]\(5\)[/tex].
Next, we need to find the slope of a line that is perpendicular to the line [tex]\(AB\)[/tex]. The property of perpendicular lines states that the slopes of two perpendicular lines are negative reciprocals of each other.
If the slope of the given line is [tex]\(m\)[/tex], the slope of a line that is perpendicular to it will be [tex]\(-\frac{1}{m}\)[/tex].
Given that the slope of the line [tex]\(AB\)[/tex] is [tex]\(5\)[/tex], the slope of the perpendicular line is:
[tex]\[
-\frac{1}{5}
\][/tex]
Therefore, the slope of a line perpendicular to the line [tex]\(AB\)[/tex] is [tex]\(\boxed{-0.2}\)[/tex].
