Question

Which points are on the perpendicular bisector of the given segment? Check all that apply. Please explain how you got your answers
(−8, 19)
(1, −8)
(0, 19)
(−5, 10)
(2, −7)

Answer 1

first, you have to find the equation of the perpendicular bisector of this given line. 
to do that, you need the slope of the perpendicular line and one point.
Step 1: find the slope of the given line segment. We have the two end points (10, 15) and (-20, 5), so the slope is m=(15-5)/(10-(-20))=1/3
the slope of the perpendicular line is the negative reciprocal of the slope of the given line, m=-3/1=-3 
step 2: find the middle point: x=(-20+10)/2=-5, y=(15+5)/2=10      (-5, 10)
so the equation of the perpendicular line in point-slope form is (y-10)=-3(x+5)

now plug in all the given coordinates to the equation to see which pair fits:
(-8, 19): 19-10=9, -3(-8+5)=9, so yes, (-8, 19) is on the perpendicular line. 

try the other pairs, you will find that (1,-8) and (-5, 10) fit the equation too. (-5,10) happens to be the midpoint. 

Answer 2

The points that are on the perpendicular bisector of the given segment is (-5, 10)

The standard equation of a line in point-slope form is expressed as

y-y0 = m(x-x0)

First, we need to get the equation of the line using the coordinates (-20, 0) and (10, 15)

Get the slope of the line

m = 15-0/10-(-20)

m = 15/30

m = 1/2

The slope of the line perpendicular to the line is -2

Using the point (-5, 10) on the line, get the equation in the point-slope form:

y - 10 = -2(x + 5)

y - 10 = -2x - 10

y + 2x = -10 + 10

y + 2x = 0

y = -2x

To check which points are on the perpendicular bisector of the given segment, hence;

For the coordinate (-8, 19)

19 = -2(-8)

19 ≠ 16 (This is not a solution)

For the coordinate (1, -8)

-8 = -2(1)

-8 ≠ -2 (This is not a solution)

For the coordinate (-5, 10)

10 = -2(-5)

10 = 10 (This is a solution)

Hence the point that is on the perpendicular bisector of the given segment is (-5, 10)

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