Step-by-step Solution:

The projections of a line segment onto the X, Y, and Z coordinate axes are equivalent to the direction ratios (a, b, c) of that line segment. We can use these values to find both the length of the segment and its direction cosines.

1. Calculate the Length of the Line Segment:
The given projections are 12, 4, and 3. These are the direction ratios:
\(a = 12, b = 4, c = 3\)
The length of the line segment, \(L\), is the magnitude of the vector represented by these direction ratios. We can find it using the 3D distance formula:
\[L = \sqrt{a^2 + b^2 + c^2}\]
Substituting the given values:
\[\begin{aligned} L &= \sqrt{12^2 + 4^2 + 3^2} \\ &= \sqrt{144 + 16 + 9} \\ &= \sqrt{169} \\ &= 13 \end{aligned}\]
Thus, the length of the line segment is 13 units.

2. Calculate the Direction Cosines:
The direction cosines (denoted as l, m, n) represent the cosines of the angles the line segment makes with the X, Y, and Z axes, respectively. They are found by dividing each direction ratio by the length (magnitude) of the segment.
\[l = \frac{a}{L}, \quad m = \frac{b}{L}, \quad n = \frac{c}{L}\]
Substituting our values:
\[l = \frac{12}{13}\]\[m = \frac{4}{13}\]\[n = \frac{3}{13}\]
So, the direction cosines of the line segment are \(\frac{12}{13}, \frac{4}{13}, \frac{3}{13}\).

Result:
The length of the line segment is 13, and its direction cosines are \(\frac{12}{13}, \frac{4}{13}, \frac{3}{13}\).