# Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints A(1,4) and B(-5, -2).

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Let the line be :

( y- y1) = m (x-x1) where (x1,x2) is any point on the line.

We know that the line pass through the midpoint of the segment AB"

==> Let M be the mid point:

M = ( x1+x2)/2 , (y1+ y2)/2

= (-5+1/2 , (-2 + 4)/2

= ( -2, 1)

Then the point M (-2,1) passes through the line.

Then:

( y - 1) = m1 (x+2)

Now we will calculate the slope

m2= ( -2-4)/(-5-1) = -6/-6 = 1

Then we know that the products of the slopes = -1

Then m1*m2 = -1

==> m1*1 = -1

==> m1= -1

==> ( y- 1) = -1(x+2)

==> y-1 = -x -2

==> y= -x -2 + 1

**==> y= -x -1 is a perpendicular line passes through bisector of AB **

The equation of the line in point slope form is :

y-y1 = m(x-x1) which has a slope m amd passes through the point (x1,y1).

A(1,4) and B (-5, -2) has the mid point M (x,y) given by:

M(x,y) = ( (Ax +Bx)/2 , (Ay+By)/2 ) =

Mx = (1-5)/2 = -2.

My = (4-2)/2 = 1.

Therefore M(x,y) = M(-2 , 1).

Slope of AB = m = (By - Ay)/(Bx- Ax) = (-2-4)/(-5-1) = 1.

Therefore the slope a line perpendicular to AB = -1/m = -1/1 = -1.

Therfore the equation of the line with slope -1 and passing through the mid point M(-2 , 1) of A and B in point slope form is given by:

**y -1 = (-1)(x-(-2)**). Or

**y-1 = -1 (x+2).**