1. **Problem Statement:** In triangle $\triangle DEF$, point $C$ is the centroid. Given that $HD = 30x - 6y$, find the expression for $CD$. 2. **Key Concept:** The centroid of a triangle divides each median into a ratio of 2:1, with the longer segment being from the vertex to the centroid. 3. **Explanation:** Since $C$ is the centroid, and $HD$ is a segment from $H$ to $D$ on the median, the segment $CD$ is exactly \textbf{one-third} of the entire median $HD$. 4. **Formula:** $$CD = \frac{1}{3} HD$$ 5. **Calculation:** Given $HD = 30x - 6y$, then $$CD = \frac{1}{3} (30x - 6y) = 10x - 2y$$ 6. **Answer:** The expression representing $CD$ is $10x - 2y$, which corresponds to option D.