Answer:
[tex]3x+y=0[/tex]
Step-by-step explanation:
step 1
Find the slope of segment HI
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
H (-4,2), and I (2,4)
substitute the given points
[tex]m=\frac{4-2}{2+4}[/tex]
[tex]m=\frac{2}{6}[/tex]
simplify
[tex]m=\frac{1}{3}[/tex]
step 2
Find the slope of the perpendicular line to segment HI
we know that
If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)
[tex]m_1*m_2=-1[/tex]
we have
[tex]m_1=\frac{1}{3}[/tex]
so
[tex]m_2=-3[/tex]
step 3
Find the midpoint segment HI
we know that
The formula to calculate the midpoint between two points is equal to
[tex]M(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]
we have
H (-4,2), and I (2,4)
substitute
[tex]M(\frac{-4+2}{2},\frac{2+4}{2})[/tex]
[tex]M(-1,3)[/tex]
step 4
we know that
The perpendicular bisector of HI is a line perpendicular to HI that passes though the midpoint of HI
Find the equation of the perpendicular bisector of HI in point slope form
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=-3[/tex]
[tex]M(-1,3)[/tex]
substitute
[tex]y-3=-3(x+1)[/tex]
step 5
Convert to slope intercept form
[tex]y=mx+b[/tex]
Isolate the variable y
[tex]y-3=-3x-3[/tex]
[tex]y=-3x-3+3[/tex]
[tex]y=-3x[/tex]
step 6
Convert to standard form
[tex]Ax+By=C[/tex]
where
A is a positive integer
B and C are integers
[tex]y=-3x[/tex]
Adds 3x both sides
[tex]3x+y=0[/tex]
see the attached figure to better understand the problem
