1. **State the problem:** Given triangle TUV with circumcenter Z, and segments XU=19, ZW=15, ZV=21, and VU=34, find the missing segment lengths WV and TV. 2. **Recall properties of a circumcenter:** The circumcenter is equidistant from all vertices of the triangle. This means: $$ ZT = ZU = ZV $$ 3. **Given:** - $ZU = 21$ (from the table) - $ZV = 21$ (given) So, $ZT = 21$ as well. 4. **Use the given segment lengths:** - $ZW = 15$ - $ZV = 21$ Since $W$ lies on side $TV$, and $Z$ lies inside the triangle, segment $TV$ can be expressed as: $$ TV = TW + WV $$ 5. **Find $TW$:** Since $Z$ lies on segment $TW$ (or near it), and $ZW=15$, and $ZT=21$, then: $$ TW = TZ - ZW = 21 - 15 = 6 $$ 6. **Find $WV$:** Given $TV$ is unknown, but $VU=34$ is given. Since $VY=17$ is given and correct, and $Y$ lies on $UV$, but not directly related to $WV$, we focus on $TV$. 7. **Calculate $TV$:** Using the triangle side $TV$, which is the sum of $TW$ and $WV$: $$ TV = TW + WV = 6 + WV $$ 8. **Use the triangle inequality or given data:** Since $TU=38$ and $VU=34$, and $TV$ is missing, but no direct value is given, we can estimate $TV$ using the Law of Cosines or given data. 9. **Since $WV$ is missing, and $TV$ is missing, but $WV$ lies on $TV$, and $TW=6$, we can find $WV$ if $TV$ is known.** 10. **Assuming $TV$ is the length between points $T$ and $V$, and $WV$ is the segment from $W$ to $V$, then:** $$ WV = TV - TW = TV - 6 $$ 11. **From the problem, $WV$ and $TV$ are missing, so we need to find $TV$ first.** 12. **Use the fact that $Z$ is the circumcenter, so $ZT=ZU=ZV=21$.** 13. **Use the triangle side $TU=38$ and $VU=34$ to find $TV$ using the Law of Cosines:** Let $TV = x$. Using the Law of Cosines on triangle $TUV$: $$ TU^2 = TV^2 + VU^2 - 2 \cdot TV \cdot VU \cdot \cos(\angle TVU) $$ But we don't have the angle, so instead, use the fact that $Z$ is the circumcenter and distances from $Z$ to vertices are equal. 14. **Alternatively, since $Z$ is the circumcenter, the distances from $Z$ to vertices are equal, and $ZW=15$ lies on $TV$, then $WV = ZV - ZW = 21 - 15 = 6$.** 15. **Therefore, $WV = 6$.** 16. **Calculate $TV$:** $$ TV = TW + WV = 6 + 6 = 12 $$ 17. **Final answers:** - $WV = 6$ - $TV = 12$ **Summary:** - $WV = 6$ - $TV = 12$