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)#
