So the problem of finding the minimum value of EF is the problem of finding the minimum value of PC. Obviously, when PC is the height on the hypotenuse of △ABC, PC is the smallest, so EF is the smallest.

As can be seen from the figure, PFEC is a rectangle with the diagonal EF=PC.

That is, find the minimum value of PC AB = 1, BC/AC = 3/4, BC = 3/5 AC = 4/5.

When PC is the smallest, that is, PC is perpendicular to AB, then PC is the smallest, that is, the height on the hypotenuse of triangle.

It can be calculated by the area method as AC*BC/AB= 12/25.

The minimum EF is equal to 12/25.

12 times the root sign of 7 times 2.