Question 1125712
Write the equation of the perpendicular bisector of the line segment with endpoints 
({{{-4}}},{{{1}}}) and ({{{4}}},{{{-3}}}) 

 first find the equation of the line containing the segment with endpoints at  ({{{-4}}},{{{1}}}) and ({{{4}}},{{{-3}}}) :

{{{y=mx+b}}}

use given points to find a slope: 

{{{m=(y[1]-y[2])/(x[1]-x[2])}}}

{{{m=(1-(-3))/(-4-4)}}}
{{{m=(1+3)/(-4-4)}}}
{{{m= 4/(-8)}}}
{{{m= -1/2}}}

{{{y=-(1/2)x+b}}}

use one point to find {{{b}}}

{{{y=-(1/2)x+b}}}...........({{{-4}}},{{{1}}}) 

{{{1=-(1/2)(-4)+b}}}
{{{1= 2+b}}}
{{{1-2=b}}}
{{{b=-1}}}

equation is: {{{y=-(1/2)x-1}}}

now, recall:
A bisector cuts a line segment into two congruent parts. A segment bisector is called a perpendicular bisector when the bisector intersects the segment at a right angle.

the perpendicular bisector  passes through the midpoint

so, first find  the coordinates of the midpoint:

({{{(-4+4)/2}}},{{{(1+(-3))/2}}})

({{{0}}},{{{-1}}})

since bisector perpendicular to line segment, it is also perpendicular to line {{{y=-(1/2)x-1}}}

and perpendicular lines have slopes negative reciprocal to each other

so, the perpendicular bisector will have a slope {{{m[p]=-1/m}}}
{{{m[p]=(-1)/(-1/2)}}}


 {{{m[p]= 2}}}

{{{y= 2x+b}}}....use midpoint ({{{0}}},{{{-1}}}) to find {{{b}}}

{{{-1= 2*0+b}}}

{{{b=-1}}}

{{{highlight(y= 2x-1)}}}->the equation of the perpendicular bisector 


{{{drawing( 600, 600, -10, 10, -10, 10,
circle(-4,1,.12),circle(4,-3,.12),line(-4,1,4,-3),
locate(-4,1,p(-4,1)),locate(4,-3,p(4,-3)),locate(0.3,-1,M(0,-1)),
 graph( 600, 600, -10, 10, -10, 10, -(1/2)x-1, 2x-1)) }}}