The medians of 

△*TUV*

 are 

*TX*

, 

*UY*

, and 

*VW*

. They meet at a single point 

*Z*

.
(In other words, 

*Z*

 is the centroid of 

△*TUV*

.)
Suppose 

=*UY*33

, 

=*TZ*8

, and 

=*VZ*14

.
Find the following lengths.
Note that the figure is not drawn to scale.


*T*

*U*

*V*

*W*

*X*

*Y*

*Z*

=*ZW*

=*ZY*

=*TX*


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. \( TX = 12 \).

Would you like a detailed geometric explanation or help with any related questions? 

---

### 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.

The medians of △TUV are TX, UY, and VW. They meet at a single point Z (the centroid of △TUV). Suppose UY = 33, TZ = 8, and VZ = 14. Find the following lengths: ZW, ZY, TX.

Explore how to calculate lengths of medians in a triangle when the centroid divides them in a 2:1 ratio. Solve for ZW, ZY, and TX in △TUV with medians TX, UY, VW, and centroid Z. Given UY = 33, TZ = 8, and VZ = 14, this step-by-step guide is perfect for high school geometry students.

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