The formula to find the mid-point of a line segment give the two end points is:

#M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)#

Where #M# is the midpoint and the given points are:

#(color(red)((x_1, y_1)))# and #(color(blue)((x_2, y_2)))#

Substituting the information we have from the problem gives:

#(8, 2) = ((color(red)(5) + color(blue)(x_2))/2 , (color(red)(6) + color(blue)(y_2))/2)#

First, we can solve for #x_2#:

#8 = (5 + x_2)/2#

#color(red)(2) xx 8 = color(red)(2) xx (5 + x_2)/2#

#16 = cancel(color(red)(2)) xx (5 + x_2)/color(red)(cancel(color(black)(2)))#

#16 = 5 + x_2#

#-color(red)(5) + 16 = -color(red)(5) + 5 + x_2#

#11 = 0 + x_2#

#11 = x_2#

Next, we can solve for #y_2#

#2 = (6 + y_2)/2#

#color(red)(2) xx 2 = color(red)(2) xx (6 + y_2)/2#

#4 = cancel(color(red)(2)) xx (6 + y_2)/color(red)(cancel(color(black)(2)))#

#-color(red)(6) + 4 = -color(red)(6) + 6 + y_2#

#-2 = 0 + y_2#

#-2 = y_2#

The other end point is #(11, -2)#