Well, the question lacks information, because there is an infinite number of solutions, and remember that a square is a hombus, and hence, your solution is one of the many which exists.
The line through AB has equation 12y = -5x + 130
The midpoint AB is (8, 7.5)
The line perpendicular to AB through (8, 7.5) has equation: 10y = 24x - 117
So basically, any pair of points equidistant from (12, 7.5) and on the line 10y = 24x - 117 is a solution.
Since your points satisfy this, they are correct
Unless more information are given.