1. **Problem Statement:** We have triangle $\triangle TUV$ with circumcenter $D$, where $\overline{AD}$, $\overline{BD}$, and $\overline{CD}$ are perpendicular bisectors of the sides. Given: $CD=30$, $BV=74$, and $UD=78$. We need to find $UV$, $VD$, and $TC$. 2. **Key Concept:** The circumcenter $D$ is equidistant from all vertices of the triangle. This means: $$DT = DU = DV = R$$ where $R$ is the circumradius. 3. **Using the given lengths:** - Since $D$ lies on the perpendicular bisector of $UV$, $D$ is equidistant from $U$ and $V$, so $DU = DV$. - Given $UD = 78$, so $DV = 78$. 4. **Find $UV$:** Since $B$ is the midpoint of $UV$ (because $BD$ is the perpendicular bisector), $$UV = 2 \times BV = 2 \times 74 = 148$$ 5. **Find $TC$:** Similarly, $C$ is the midpoint of $TU$ (since $CD$ is the perpendicular bisector), so $$TC = CU$$ Since $D$ is the circumcenter, $DT = DU = DV = R$. We know $CD = 30$, and $C$ is midpoint of $TU$, so $TC = CU$. 6. **Find $TC$ using the circumradius:** Since $DT = DU = DV = R$, and $CD = 30$ is the segment from $C$ to $D$ on the perpendicular bisector of $TU$, the length $TC$ is twice the distance from $C$ to $T$ (since $C$ is midpoint), but we need more info to find $TC$ directly. However, since $D$ is the circumcenter, $DT = DU = DV = R$. We have $DU = 78$, so $R = 78$. 7. **Summary of answers:** - $UV = 148$ - $VD = 78$ - $TC$ is unknown from given data but since $C$ is midpoint of $TU$, and $D$ lies on perpendicular bisector, $TC$ is equal to $CU$, but no length given for $TU$ or $TC$. Assuming $TC = CD = 30$ (since $C$ lies on side $TU$ and $CD$ is perpendicular bisector segment), then $$TC = 30$$ **Final answers:** $$UV = 148, \quad VD = 78, \quad TC = 30$$