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In triangle \( \triangle PQR \), point \( V \) is the centroid, which is the intersection of the three medians. The centroid divides each median into two segments, with the ratio \( 2:1 \), meaning that the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.

Here, we are given:
- \( PT = 18 \), where \( T \) is the midpoint of side \( QR \) (since \( PT \) is a median of the triangle).
- We need to find \( VT \).

Since \( V \) divides \( PT \) in the ratio \( 2:1 \), we have:
\[
PT = PV + VT
\]
where \( PV : VT = 2:1 \).

Thus, we can write:
\[
PT = 3 \times VT
\]
Substituting \( PT = 18 \):
\[
18 = 3 \times VT
\]
\[
VT = \frac{18}{3} = 6
\]

### Answer:
\[
VT = 6
\]

Would you like more details on the solution or have any further questions?

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### Related Questions:
1. How does the centroid divide each median in a triangle?
2. What is the formula for finding the centroid of a triangle with given vertices?
3. How can we prove that the centroid divides each median in the \( 2:1 \) ratio?
4. How does this concept apply if the triangle is not isosceles?
5. What other properties does the centroid of a triangle have?

### Tip:
In a triangle, the centroid is always located inside the triangle and serves as its center of mass, balancing each median at a \( 2:1 \) ratio from the vertex.

In \( \triangle PQR \), \( V \) is the intersection of the three medians. If \( PT = 18 \), find \( VT \).

Solve for \( VT \) in triangle \( \triangle PQR \) where the centroid divides the median \( PT = 18 \) in a 2:1 ratio. This problem covers geometry concepts of centroid, median, and ratios, suitable for Grades 9-10.

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