1. **Problem Statement:** We have a circle with center $P$ and diameter $\overline{AB}$. Points $A$, $B$, $C$, and $D$ lie on the circle in clockwise order. We know the central angles $\angle APD = 7x + 1$ degrees and $\angle BPC = 9x - 7$ degrees. We need to find the measure of the major arc $\stackrel{\large{\frown}}{ACD}$ in degrees. 2. **Key facts and formulas:** - The diameter $\overline{AB}$ divides the circle into two semicircles, each measuring 180 degrees. - The sum of all central angles around point $P$ is 360 degrees. - The major arc $\stackrel{\large{\frown}}{ACD}$ is the arc from $A$ to $D$ passing through $C$, which is the complement of the minor arc $\stackrel{\large{\frown}}{ABD}$. 3. **Express the known angles:** - $\angle APD = 7x + 1$ - $\angle BPC = 9x - 7$ 4. **Find the remaining central angles:** - Since $\overline{AB}$ is a diameter, $\angle APB = 180^\circ$. - The circle is divided into arcs $\stackrel{\large{\frown}}{AB}$ and $\stackrel{\large{\frown}}{BA}$, each 180 degrees. 5. **Sum of central angles around $P$:** $$\angle APD + \angle BPC + \angle CPD + \angle BPA = 360^\circ$$ 6. **Identify arcs and angles:** - $\angle APD$ corresponds to arc $\stackrel{\large{\frown}}{AD}$. - $\angle BPC$ corresponds to arc $\stackrel{\large{\frown}}{BC}$. 7. **Since $\overline{AB}$ is diameter, arcs $\stackrel{\large{\frown}}{AB}$ and $\stackrel{\large{\frown}}{BA}$ are 180 degrees each. The arcs $\stackrel{\large{\frown}}{ACD}$ and $\stackrel{\large{\frown}}{B}$ are complementary to these. We want $\stackrel{\large{\frown}}{ACD}$, which is the major arc from $A$ to $D$ passing through $C$. 8. **Calculate $x$ by using the fact that $\angle APD$ and $\angle BPC$ are parts of the circle:** - The sum of arcs $\stackrel{\large{\frown}}{AD}$ and $\stackrel{\large{\frown}}{BC}$ plus the arcs $\stackrel{\large{\frown}}{CD}$ and $\stackrel{\large{\frown}}{AB}$ equals 360 degrees. 9. **Since $\overline{AB}$ is diameter, $\stackrel{\large{\frown}}{AB} = 180^\circ$. So the sum of $\angle APD$ and $\angle BPC$ plus the remaining arcs equals 360 degrees. But $\angle APD + \angle BPC$ are given in terms of $x$. 10. **Sum the given angles:** $$ (7x + 1) + (9x - 7) = 16x - 6 $$ 11. **Since $\angle APD$ and $\angle BPC$ are central angles corresponding to arcs $AD$ and $BC$, and the diameter divides the circle into two 180 degree arcs, the sum of these two arcs plus the other arcs must be 360 degrees. The other arcs are $\stackrel{\large{\frown}}{CD}$ and $\stackrel{\large{\frown}}{BA}$, but $\stackrel{\large{\frown}}{BA} = 180^\circ$ (diameter arc). So the sum of $\angle APD + \angle BPC + 180 = 360$. 12. **Set up the equation:** $$ 16x - 6 + 180 = 360 $$ 13. **Solve for $x$:** $$ 16x + 174 = 360 $$ $$ 16x = 186 $$ $$ x = \frac{186}{16} = 11.625 $$ 14. **Calculate $\angle APD$ and $\angle BPC$ using $x$:** $$ \angle APD = 7(11.625) + 1 = 81.375 + 1 = 82.375^\circ $$ $$ \angle BPC = 9(11.625) - 7 = 104.625 - 7 = 97.625^\circ $$ 15. **Find the minor arc $\stackrel{\large{\frown}}{ACD}$:** - The minor arc $\stackrel{\large{\frown}}{ACD}$ corresponds to the sum of $\angle APD$ and $\angle BPC$ because these central angles subtend arcs $AD$ and $BC$ respectively. 16. **Calculate the minor arc $\stackrel{\large{\frown}}{ACD}$:** $$ 82.375^\circ + 97.625^\circ = 180^\circ $$ 17. **Since the circle is 360 degrees, the major arc $\stackrel{\large{\frown}}{ACD}$ is:** $$ 360^\circ - 180^\circ = 180^\circ $$ **Final answer:** The measure of the major arc $\stackrel{\large{\frown}}{ACD}$ is $180$ degrees.