Answer:
[tex]y=\frac{1}{2}x-3[/tex]
Step-by-step explanation:
step 1
Find the slope of AB
we know that
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
A(5,-1), B(9, -9)
substitute the values
[tex]m=\frac{-9+1}{9-5}[/tex]
[tex]m=\frac{-8}{4}[/tex]
[tex]m=-2[/tex]
step 2
Find the slope of the line that is perpendicular to AB
we know that
If two lines are perpendicular, their their slopes are opposite reciprocal (the product of their slopes is equal to -1)
we have
[tex]m_A_B=-2[/tex]
therefore
The slope of the perpendicular line is
[tex]m=\frac{1}{2}[/tex]
step 3
Find the equation of the line in point slope form
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=\frac{1}{2}[/tex]
[tex]C(4,-1)[/tex]
substitute
[tex]y+1=\frac{1}{2}(x-4)[/tex]
step 4
Convert to slope intercept form
Isolate the variable y
[tex]y+1=\frac{1}{2}x-2\\\\y=\frac{1}{2}x-2-1\\\\y=\frac{1}{2}x-3[/tex]
