To find the possible x-coordinate of the fourth vertex of a parallelogram, we can use the fact that opposite sides of a parallelogram are parallel.

Given three vertices of the parallelogram: (1, 2), (3, 8), and (4, 1), we need to find the x-coordinate of the fourth vertex.

Let's denote the fourth vertex as (x, y).

Identifying the Opposite Side
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In a parallelogram, the opposite sides are parallel. Therefore, the line connecting the first two given vertices: (1, 2) and (3, 8), is parallel to the line connecting the third given vertex: (4, 1), and the fourth vertex: (x, y).

Finding the equation of the line connecting (1, 2) and (3, 8)
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We can use the two-point form of a line to find the equation of the line connecting (1, 2) and (3, 8).

The slope of the line is given by:
m = (y2 - y1) / (x2 - x1)
= (8 - 2) / (3 - 1)
= 6 / 2
= 3

Using the point-slope form of a line (y - y1) = m(x - x1), we can substitute one of the given points, say (1, 2), into the equation to find the equation of the line:
(y - 2) = 3(x - 1)

Simplifying the equation, we get:
y - 2 = 3x - 3
y = 3x - 1

Finding the equation of the line connecting (4, 1) and (x, y)
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Since the line connecting (4, 1) and the fourth vertex (x, y) is parallel to the line connecting (1, 2) and (3, 8), it will have the same slope.

Using the two-point form of a line, we can find the equation of the line connecting (4, 1) and (x, y):
(y - 1) = 3(x - 4)

Simplifying the equation, we get:
y = 3x - 11

Finding the x-coordinate of the fourth vertex
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Since the x-coordinate of the fourth vertex satisfies both equations (y = 3x - 1 and y = 3x - 11), we can set the two equations equal to each other and solve for x:

3x - 1 = 3x - 11

Simplifying the equation, we get:
-1 = -11

Since this equation is not true, there is no x-coordinate that satisfies both equations simultaneously. Therefore, the fourth vertex cannot exist, and the sum of all possible x-coordinates of the fourth vertex is 0.

Therefore, the correct answer is option 'D' (0).