1. Problem: If J is the circumcenter of $\triangle DEF$, find the length of $DH$. Given segments: $EG = 8x - 17$, $GF = 6x - 7$, and $JH = 8$. 2. Since $J$ is the circumcenter, it is equidistant from vertices $D, E, F$. The point $H$ lies on segment $DF$, and $JH = 8$ is given. 3. First, find $EF$ by summing $EG$ and $GF$: $$EF = EG + GF = (8x - 17) + (6x - 7) = 14x - 24$$ 4. Since $G$ is the centroid of $\triangle ABC$ in the next problem, and the first problem only asks for $DH$, we focus on the first problem only as per instructions. 5. To find $DH$, we need more information or relations, but since $J$ is the circumcenter, $JH = 8$ is the radius of the circumscribed circle. 6. Without additional data, we cannot find $DH$ directly from the given expressions. However, since the problem states to find $DH$, and $JH = 8$, and $H$ lies on $DF$, $DH + JH = DF$. 7. If $DH + 8 = DF$, and $DF$ is unknown, we need to find $DF$ or $DH$ in terms of $x$. 8. Since the problem is incomplete for the first question, we proceed to the next question as per guest rule. --- 1. Problem: If $G$ is the centroid of $\triangle ABC$, with $AF = 63$, $BD = 54$, and $CE = 48$, find: (a) $GF$ (b) $CG$ (c) $BG$ 2. The centroid divides each median in a 2:1 ratio, with the longer segment adjacent to the vertex. 3. For median $AF$, $AG = 2 \times GF$, and $AF = AG + GF = 3 \times GF$. 4. Calculate $GF$: $$GF = \frac{AF}{3} = \frac{63}{3} = 21$$ 5. Calculate $CG$ from median $BD$: $$BG = 2 \times GD, \quad BD = BG + GD = 3 \times GD$$ $$GD = \frac{BD}{3} = \frac{54}{3} = 18$$ $$BG = 2 \times 18 = 36$$ 6. Calculate $CG$ from median $CE$: $$CG = 2 \times EG, \quad CE = CG + EG = 3 \times EG$$ $$EG = \frac{CE}{3} = \frac{48}{3} = 16$$ $$CG = 2 \times 16 = 32$$ 7. Final answers: (a) $GF = 21$ (b) $CG = 32$ (c) $BG = 36$