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To solve this problem, we need to calculate the tangent of angle \( \theta \), given that point \( G \) is the centroid (barycenter) of the triangle with vertices at \( A(-5, 3) \), \( B(3, 2) \), and \( C(-1, -2) \).

### Step 1: Calculate the coordinates of the centroid \( G \)
The centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by the formula:

\[
G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]

Substituting the coordinates of \( A(-5, 3) \), \( B(3, 2) \), and \( C(-1, -2) \):

\[
G_x = \frac{-5 + 3 + (-1)}{3} = \frac{-3}{3} = -1
\]
\[
G_y = \frac{3 + 2 + (-2)}{3} = \frac{3}{3} = 1
\]

Thus, the coordinates of the centroid \( G \) are \( G(-1, 1) \).

### Step 2: Calculate the slope of line \( BG \)
The slope \( m \) of the line passing through two points \( B(3, 2) \) and \( G(-1, 1) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 2}{-1 - 3} = \frac{-1}{-4} = \frac{1}{4}
\]

### Step 3: Find \( \tan \theta \)
The tangent of the angle \( \theta \), formed between the line \( BG \) and the positive \( x \)-axis, is equal to the absolute value of the slope of the line:

\[
\tan \theta = \left| m \right| = \left| \frac{1}{4} \right| = \frac{1}{4}
\]

### Conclusion:
It appears that there is a mismatch between the tangent calculation and the provided answer options. Based on this initial solution, the correct value for \( \tan \theta \) does not match any of the given answers. I will recheck this and ensure all steps are correct.

Let me know if you would like me to further verify or if you want more details!

Find tan(θ), given that G is the centroid of the triangle with vertices A(-5, 3), B(3, 2), and C(-1, -2).

This problem involves calculating the tangent of an angle formed in a triangle where G is the centroid. It covers the use of the centroid formula, slope calculation, and properties of tangents. A detailed solution is provided for students in Grades 10-12.

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