Given:

#x_"starting point" = 0#,

#y_"starting point" = 0#

#x_"midpoint" = 2#

#y_"midpoint" = -8#

Find: #x_"other end"# and #y_"other end"#

The change in x, #Deltax#, from the starting point to the midpoint is:

#Deltax = x_"midpoint" - x_"starting point"" [1]"#

The other end must be twice that change relative to the starting point:

#x_"other end" = 2Deltax+ x_"starting point"" [2]"#

Substitute the right side of equation [1] into equation [2]:

#x_"other end" = 2(x_"midpoint" - x_"starting point")+ x_"starting point"" [3]"#

Use the distributive property:

#x_"other end" = 2x_"midpoint" - 2x_"starting point" + x_"starting point"" [4]"#

Combine like terms:

#x_"other end" = 2x_"midpoint" - x_"starting point"" [5]"#

The same thing is true of the y coordinate:

#y_"other end" = 2y_"midpoint" - y_"starting point"" [6]"#

Substituting the given information into equations [5] and [6]:

#x_"other end" = 2(2) - 0#

#y_"other end" = 2(-8) - 0#

#x_"other end" = 4#

#y_"other end" = -16#

The other endpoint is #(4,-16)#