To solve this problem, we need to understand that in a triangle, the centroid $O$ divides each median into a ratio of $2:1$. This implies that the longer segment from a vertex to the centroid is twice the length of the shorter segment from the centroid to the midpoint of the opposite side.

In this case:

1. Given that $O$ is the centroid, we can set up a relationship between the expressions given on the triangle's sides, specifically the segment lengths $2x - 7$ and $3x$.

2. Since the centroid divides each median in a $2:1$ ratio, we set up the equation:

   $2x - 7 = 2 \cdot (3x)$

3. Simplify this equation:

   $2x - 7 = 6x$

4. Rearranging terms to isolate $x$:

   $-7 = 4x$

5. Solving for $x$:

   $x = -\frac{7}{4}$

Thus, none of the options directly match this answer, so double-checking the approach or if there might be an alternative interpretation could be necessary.

Let me know if you need more detailed reasoning or have any questions.

Related Questions:

1. How is the centroid of a triangle defined in terms of medians?
2. Can you explain the $2:1$ ratio property of a centroid more thoroughly?
3. How does one approach centroid problems involving variable expressions?
4. Are there other ways to solve this problem using vector or coordinate geometry?
5. How can you verify if the calculated value satisfies the given conditions?

Tip: When dealing with centroids, always remember that they split medians into a $2:1$ ratio from the vertex to the midpoint of the opposite side.