1. **State the problem:** We have a circle with center at point $P$ and diameter $\overline{AB}$. Points $A$, $B$, $C$, and $D$ lie on the circle in clockwise order. Given the angles $\angle APD = 7x + 1$ degrees and $\angle BPC = 9x - 7$ degrees, we need to find the arc measure of the major arc $\stackrel{\large{\frown}}{ACD}$. 2. **Analyze given information:** Since $\overline{AB}$ is a diameter, $A$ and $B$ lie at opposite ends of the circle, so arc $\stackrel{\large{\frown}}{AB}$ is $180^\circ$. 3. **Recall important properties:** - Angle $\angle APD$ is the central angle intercepting arc $\stackrel{\large{\frown}}{AD}$. - Angle $\angle BPC$ is the central angle intercepting arc $\stackrel{\large{\frown}}{BC}$. - The sum of angles around point $P$ is $360^\circ$, so: $$\angle APD + \angle BPC + \angle CP A + \angle APB = 360^\circ$$ But since $A, B, C, D$ lie on circle and $A,B$ are diameter ends, we consider these two central angles corresponding to arcs $AD$ and $BC$ respectively. 4. **Key step: Sum of arcs:** We know: $$\text{arc } AD + \text{arc } BC = 360^\circ - \text{arc } AB = 360^\circ - 180^\circ = 180^\circ$$ Because $AB$ is diameter, it subtends a $180^\circ$ arc. 5. **Find $x$ using the angles at center:** Central angles equal their arcs, so: $$7x + 1 + 9x - 7 = 180$$ $$16x - 6 = 180$$ $$16x = 186$$ $$x = \frac{186}{16} = 11.625$$ 6. **Calculate the central angles in degrees:** $$\angle APD = 7(11.625) + 1 = 81.375 + 1 = 82.375^\circ$$ $$\angle BPC = 9(11.625) - 7 = 104.625 - 7 = 97.625^\circ$$ 7. **Determine arcs:** - Arc $AD$ corresponds to $82.375^\circ$ (angle $APD$) - Arc $BC$ corresponds to $97.625^\circ$ (angle $BPC$) 8. **Find arc $ACD$ (major arc):** Note the circle goes $A \to C \to D \to B$ in clockwise order. - Arc $ACD$ consists of arcs $AC + CD$. - Since the arcs $AD + BC = 180^\circ$, and $AB=180^\circ$, the major arc $ACD$ is the complement of arc $B$. But to be precise, since $AB$ is diameter and arcs $AD + BC = 180^\circ$, the major arc $ACD$ equals $360^\circ - 97.625^\circ = 262.375^\circ$. **Final answer:** $$\boxed{262.375^\circ}$$