The gradient of the perpendicular is the negative reciprocal of the gradient of the line joining the two points, which is (8-6)/(-4-4)=-2/8=-1/4 (difference of y coord divided by the difference of the x coord). Therefore, the gradient of the perpendicular is 4. The midway point is the average of the coords: ((-4+4)/2,(8+6)/2)=(0,7). The equation of the perpendicular is (y-7)=4(x-0)=4x.
[The equation of the line connecting the two points is (y-6)=-(x-4)/4 or (y-8)=-(x+4)/4. This can be written 4y-24=-x+4 or 4y=28-x; or 4y-32=-x-4 or 4y=28-x. The midway point (0,7) clearly lies on this line, and the line and perpendicular intersect when (28-x)/4=4x+7; 28-x=16x+28, i.e., when x=0 and y=7.]