1. **Problem statement:** We have triangle $\triangle TUV$ with medians $\overline{TX}$, $\overline{UY}$, and $\overline{VW}$ intersecting at centroid $Z$. Given $UY=33$, $TZ=8$, and $VZ=14$, find lengths $ZW$, $ZY$, and $TX$. 2. **Key property of centroid:** The centroid divides each median into a ratio $2:1$, with the longer segment between the vertex and centroid. 3. **Using the property on median $UY$:** Since $UY=33$, and $Z$ divides $UY$ in $2:1$ ratio, $$ UZ = \frac{2}{3} \times 33 = 22, \quad ZY = \frac{1}{3} \times 33 = 11. $$ 4. **Using the property on median $TX$:** Given $TZ=8$ and $Z$ divides $TX$ in $2:1$ ratio, $$ TZ = \frac{2}{3} TX \implies TX = \frac{3}{2} \times 8 = 12. $$ 5. **Using the property on median $VW$:** Given $VZ=14$ and $Z$ divides $VW$ in $2:1$ ratio, $$ VZ = \frac{2}{3} VW \implies VW = \frac{3}{2} \times 14 = 21. $$ 6. **Find $ZW$:** Since $ZW$ is the shorter segment, $$ ZW = \frac{1}{3} VW = \frac{1}{3} \times 21 = 7. $$ **Final answers:** $$ ZW = 7, \quad ZY = 11, \quad TX = 12. $$