Indicate the equation of the line that is the perpendicular bisector of the segment with endpoints (4, 1) and (2, -5).

Find th midopint

the mindpoint of 2 points (x1,y1) and (x2,y2) is
((x1+x2)/2,(y1+y2)/2)

so midpoint is ((4+2)/2,(1-5)/2)=(6/2,-4/2)=(3,-2)

now find the slope of the line
it is perpendicular to the line with (4,1) and (2,-5)
find the slope of th eline passing through (4,1) and (2,-5)
slope between (x1,y1) and (x2,y2) is
(y2-y1)/(x2-x1)
so the slope between (4,1) and (2,-5) is
(-5-1)/(2-4)=-6/-2=3

perpendicular lines have slopes tha tmuliplty to get -1
3 times what=-1
what=-1/3

so what is the equation of a line that passes through the point (3,-2) and has a slope of -1/3

the equation of a line that has a slope of m and passes through (x1,y1) is
y-y1=m(x-x1)
so
y-(-2)=-1/3(x-3)
y+2=-1/3(x-3)
if we want slope intercept form
y+2=-1/3x+1
y=-1/3x-1

if we want standard form
1/3x+y=-1
x+3y=-3