Shortest Distance between two lines:
The shortest distance between two skew lines can be determined by finding a perpendicular line to both lines, which passes through the point of intersection of the two lines. The distance between these two lines can be determined by finding the length of this perpendicular line.

Given Lines:
(x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 2)/3 = (y - 4)/4 = (z - 5)/5

Step 1: Writing the direction ratios of both lines.
The direction ratios of the first line are [2, 3, 4] and the direction ratios of the second line are [3, 4, 5].

Step 2: Finding the vector equation of both lines.
The vector equation of the first line is r1 = i + 2j + 3k + 2a + 3b + 4c
The vector equation of the second line is r2 = 2i + 4j + 5k + 3a + 4b + 5c

Step 3: Finding the point of intersection of both lines.
The point of intersection can be found by equating the vector equation of both lines and solving for a, b, and c.
i + 2j + 3k + 2a + 3b + 4c = 2i + 4j + 5k + 3a + 4b + 5c
a = -1/6, b = 1/6, c = 1/6
Substitute the values of a, b, and c in any of the vector equations of the lines to get the point of intersection.
The point of intersection is (-1/6, 7/6, 5/2)

Step 4: Finding the direction vector of the line perpendicular to both lines.
The direction vector of the line perpendicular to both lines can be found by taking the cross product of the direction ratios of both lines.
[2, 3, 4] x [3, 4, 5] = [-1, 2, -1]
The direction vector of the line perpendicular to both lines is [-1, 2, -1].

Step 5: Finding the shortest distance between the lines.
The shortest distance between the lines can be found by projecting the vector joining the point of intersection and any point on the first line onto the line perpendicular to both lines.
Let's take the point (1, 2, 3) on the first line.
The vector joining the point of intersection and (1, 2, 3) is [1/6, 5/6, 1/2] - [1, 2, 3] = [-5/6, -1/6, -5/2]
The projection of this vector onto the line perpendicular to both lines is given by:
|[-5/6, -1/6, -5/2] . [-1, 2, -1]/sqrt(6)| = 1/sqrt(6)
Therefore, the shortest distance between the lines is 1/sqrt(6).