Since the shortest distance from a point to a line is along the perpendicular to the line, construct the plane through the given point perpendicular to the given line. The line is given by x= 0, y= 4+ 5t, z= 1+ 2t so the vector <0, 5, 2> is in the direction of the line and so normal to any perpendicular plane. That is, the plane perpendicular to that line and through the point (4, 4, 1) is 0(x- 4)+ 5(y- 4)+ 2(z-1)= 0 or 5y+ 2z= 22.
the line crosses that plane where 5(4+ 5t- 4)+ 2(1+ 2t- 1)= 22 or 29t= 22 so t= 22/29 and x= 0, y= 4+ 110/29= 226/29, z= 1+ 44/29= 73/29. The distance from (4,4,1) to (0, 225/29, 73/29) is distance from the point to the line. It should be just what you get using Plato's formula.