The solution is:
The hot air balloon is one mile high.
The three towns are at the vertices of an equilateral triangle with a 12 mile side.
Use the 30-60-90 triangle to determine that the center of the triangle is 4√3 miles from each city. I labeled it as 12 /√3 in the diagram below.
The height of the balloon hovering above is the third side of a right triangle in which one leg is 4√3 miles and the hypotenuse is 7 miles.
So, 72 = h2 + (4√3)2
49 = h2 + 48
Therefore, h = 1 mile
Here is Ritwik Chaudhuri's excellent solution which uses trigonometry:
Click here for Colin Bowey's excellent solution