This is a right triangle.

Thus, 5 and 12 are two of its altitutes

But it is not the smallest.

The smallest is the one drawn from intersection of 5 and 12to hypotenuse.

Let h be that length.

Then, (13h)(1/2) = Area

But, Area = (1/2)(5)(12) = 30

Thus,

13h = 60

h = 60/13 = 4.61