Answer:
[tex]y=\frac{1}{3}x+\frac{5}{3}[/tex]
Step-by-step explanation:
Let
A(-5,5),B(-2,-4) ----> the given segment
step 1
Find the slope AB
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
substitute the given values
[tex]m=\frac{-4-5}{-2+5}[/tex]
[tex]m=\frac{-9}{3}[/tex]
[tex]m=-3[/tex]
step 2
we know that
If two lines are perpendicular, then their slopes are opposite reciprocal (the product of the slopes is equal to -1)
so
the slope of the perpendicular bisector is equal to
[tex]m_1*m_2=-1[/tex]
we have
[tex]m_1=-3[/tex]
substitute
[tex](-3)*m_2=-1[/tex]
[tex]m_2=\frac{1}{3}[/tex]
step 3
Find the midpoint segment AB
A(-5,5),B(-2,-4)
The formula to calculate the midpoint between two points is equal to
[tex]M(\frac{x1+x2}{2} ,\frac{y1+y2}{2})[/tex]
substitute the values
[tex]M(\frac{-5-2}{2} ,\frac{5-4}{2})[/tex]
[tex]M(-\frac{7}{2} ,\frac{1}{2})[/tex]
step 4
Find the equation of the line in point slope form
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=\frac{1}{3}[/tex]
[tex]M(-\frac{7}{2} ,\frac{1}{2})[/tex]
substitute
[tex]y-\frac{1}{2}=\frac{1}{3}(x+\frac{7}{2})[/tex]
step 5
Convert to slope intercept form
[tex]y=mx+b[/tex]
isolate the variable y
[tex]y-\frac{1}{2}=\frac{1}{3}x+\frac{7}{6}[/tex]
[tex]y=\frac{1}{3}x+\frac{7}{6}+\frac{1}{2}[/tex]
[tex]y=\frac{1}{3}x+\frac{10}{6}[/tex]
simplify
[tex]y=\frac{1}{3}x+\frac{5}{3}[/tex]
see the attached figure to better understand the problem
