The equation of line [tex]\( AB \)[/tex] is [tex]\( (y-3)=5(x-4) \)[/tex]. What is the slope of a line perpendicular to line [tex]\( AB \)[/tex]?

A. [tex]\(-5\)[/tex]
B. [tex]\(5\)[/tex]
C. [tex]\(\frac{-1}{5}\)[/tex]
D. [tex]\(\frac{1}{5}\)[/tex]



Answer :

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].