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)

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.