The centroid $Z$ of a triangle divides each median into two segments such that the ratio of the lengths is $2:1$, with the longer segment being from the vertex to the centroid.

Known Information

1. $UY = 33$ (entire median from $U$ to $Y$).
2. $TZ = 8$ (segment from $T$ to $Z$).
3. $VZ = 14$ (segment from $V$ to $Z$).

Step 1: Length of $ZW$

The median $VW$ is divided by $Z$ into $VZ$ and $ZW$, where: $VZ : ZW = 2:1.$ From the given, $VZ = 14$. Therefore, $ZW$ is: $ZW = \frac{VZ}{2} = \frac{14}{2} = 7.$

Step 2: Length of $ZY$

The median $UY$ is divided by $Z$ into $UZ$ and $ZY$, where: $UZ : ZY = 2:1.$ The total length of $UY$ is given as 33. Therefore: $UZ = \frac{2}{3} \cdot UY = \frac{2}{3} \cdot 33 = 22,$ $ZY = \frac{1}{3} \cdot UY = \frac{1}{3} \cdot 33 = 11.$

Step 3: Length of $TX$

The entire median $TX$ is divided into $TZ$ and $ZX$, where: $TZ : ZX = 2:1.$ The given length $TZ = 8$. Therefore: $ZX = \frac{TZ}{2} = \frac{8}{2} = 4,$ $TX = TZ + ZX = 8 + 4 = 12.$

Final Results

1. $ZW = 7$.
2. $ZY = 11$.
3. $TX = 12$.

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Related Questions

1. What is the total length of the median $VW$?
2. How does the centroid divide other line segments of $\triangle TUV$?
3. Can the centroid be used to find the area of $\triangle TUV$?
4. What is the relationship between the centroid and the center of gravity of a triangle?
5. How are the medians of a triangle related to its side lengths?

Tip

The centroid of a triangle always divides the medians into a $2:1$ ratio. This property is useful in various geometric problems.