Segment Length in Right Triangle

In triangle XYZ, the lengths of the sides are in the ratio 3:4:5. The perimeter of the triangle is 72 units. If the altitude from the vertex opposite the longest side divides the longest side (the side of length 5) into two segments, what is the length of the shorter segment?

Recall that the area of a triangle can be expressed in two ways: using the base and height, or Heron's formula.