Answer :
Certainly! Let's carefully work through the problem to determine the slope of a line that is perpendicular to line [tex]\( AB \)[/tex].
Firstly, we need to extract the slope from the given equation of line [tex]\( AB \)[/tex]:
[tex]\[ (y - 3) = 5(x - 4) \][/tex]
This equation is in the point-slope form, which is given by:
[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. Comparing this with the given equation, we see that:
[tex]\[ m = 5 \][/tex]
So, the slope of line [tex]\( AB \)[/tex] is [tex]\( 5 \)[/tex].
Now, for a line to be perpendicular to another line, the slope of the perpendicular line is the negative reciprocal of the original slope. For a slope [tex]\( m \)[/tex], the negative reciprocal is:
[tex]\[ \text{Perpendicular slope} = -\frac{1}{m} \][/tex]
Given that the slope of line [tex]\( AB \)[/tex] is [tex]\( 5 \)[/tex], the slope of the line perpendicular to it will be:
[tex]\[ -\frac{1}{5} \][/tex]
Therefore, the slope of the line perpendicular to line [tex]\( AB \)[/tex] is [tex]\(\boxed{-\frac{1}{5}}\)[/tex].
Firstly, we need to extract the slope from the given equation of line [tex]\( AB \)[/tex]:
[tex]\[ (y - 3) = 5(x - 4) \][/tex]
This equation is in the point-slope form, which is given by:
[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. Comparing this with the given equation, we see that:
[tex]\[ m = 5 \][/tex]
So, the slope of line [tex]\( AB \)[/tex] is [tex]\( 5 \)[/tex].
Now, for a line to be perpendicular to another line, the slope of the perpendicular line is the negative reciprocal of the original slope. For a slope [tex]\( m \)[/tex], the negative reciprocal is:
[tex]\[ \text{Perpendicular slope} = -\frac{1}{m} \][/tex]
Given that the slope of line [tex]\( AB \)[/tex] is [tex]\( 5 \)[/tex], the slope of the line perpendicular to it will be:
[tex]\[ -\frac{1}{5} \][/tex]
Therefore, the slope of the line perpendicular to line [tex]\( AB \)[/tex] is [tex]\(\boxed{-\frac{1}{5}}\)[/tex].
