To solve this problem, we need to find the value of [tex]\( x \)[/tex] such that the line passing through the points [tex]\((1,2)\)[/tex] and [tex]\((x, 5)\)[/tex] is perpendicular to a line with a slope of [tex]\(\frac{1}{3}\)[/tex].
First, let's denote the slope of the line we want to find as [tex]\( m \)[/tex].
Given that two lines are perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]. So, if the slope of the first line is [tex]\(\frac{1}{3}\)[/tex] and the slope of the second line is [tex]\( m \)[/tex], then:
[tex]\[ m \cdot \frac{1}{3} = -1 \][/tex]
Solving for [tex]\( m \)[/tex]:
[tex]\[ m = -1 \cdot 3 \][/tex]
[tex]\[ m = -3 \][/tex]
So, the slope of the line passing through the points [tex]\((1,2)\)[/tex] and [tex]\((x, 5)\)[/tex] is [tex]\(-3\)[/tex].
We use the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the known values:
[tex]\[ -3 = \frac{5 - 2}{x - 1} \][/tex]
[tex]\[ -3 = \frac{3}{x - 1} \][/tex]
Now, we solve for [tex]\( x \)[/tex]:
[tex]\[ -3 (x - 1) = 3 \][/tex]
[tex]\[ -3x + 3 = 3 \][/tex]
[tex]\[ -3x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\( 0 \)[/tex].
Therefore, the correct option is:
[tex]\[ \boxed{0} \][/tex]
