1. The problem gives a circle with center $P$ and points $A$, $B$, $C$, $D$, and $E$ on the circle in clockwise order. 2. $\overline{AC}$ and $\overline{BE}$ are diameters, so $A$ and $C$ are endpoints of one diameter, and $B$ and $E$ are endpoints of another diameter. 3. The angles at the center are expressed in terms of $w$: - $\angle APE = 4w + 4$ degrees - $\angle APB = 4w + 8$ degrees - $\angle CPD = 2w + 11$ degrees 4. Since $A$, $B$, $C$, $D$, and $E$ lie on the circle, the central angles between these points correspond to the arcs between them. 5. The entire circle measures 360 degrees. We want the measure of the minor arc $\stackrel{\large{\frown}}{DE}$. 6. Note the relationships: - $\angle APB$ spans from $A$ to $B$. - $\angle APE$ spans from $A$ to $E$. - $\angle CPD$ spans from $C$ to $D$. 7. Since $B$ and $E$ are endpoints of a diameter, the arcs $BE$ and $EB$ each measure 180 degrees. 8. $\angle APE = 4w + 4$ degrees is the angle between $A$ and $E$. Since $A$ to $E$ goes through $B$ (because points are in clockwise order), then arc $AE = \angle APE$. 9. $\angle APB = 4w + 8$ degrees is the angle between $A$ and $B$. 10. Because $\angle APE$ includes $\angle APB$, the arc from $B$ to $E$ is: $$\text{arc }BE = \text{arc }AE - \text{arc }AB = (4w + 4) - (4w + 8) = -4 \text{ degrees}$$ 11. This negative result indicates a contradiction, so instead, use the fact that $B$ and $E$ are opposite points, so arc $BE = 180$ degrees. 12. Using this, we can find $w$ by considering angles around the circle. The total sum of angles must be 360 degrees. 13. Sum arcs for $AB$, $BC$, $CD$, $DE$, and $EA$ equal 360 degrees. 14. We know: - arc $AB = \angle APB = 4w + 8$ - arc $BC$ corresponds to $\angle CPB$ (not given) - arc $CD = \angle CPD = 2w + 11$ - arc $DE = x$ (unknown) - arc $EA = (4w + 4)$ 15. Since $AC$ is diameter, arc $AC = 180$ degrees. - arc $AB + BC = 180$ degrees - so arc $BC = 180 - (4w + 8)$ 16. Since $BE$ is diameter, arc $BE = 180$ degrees. - arc $BD + DE = 180$ degrees But $BD = arc BC + arc CD = [180 - (4w + 8)] + (2w + 11) = 180 - 4w - 8 + 2w + 11 = 183 - 2w$ - Then $DE = 180 - BD = 180 - (183 - 2w) = -3 + 2w$ 17. So arc $DE = 2w - 3$ degrees. 18. Finally, to find $w$, consider angles at $P$ around the circle summing to 360 degrees: $$\angle APB + \angle CPD + \angle DPE + \angle EPA = 360$$ But $\angle DPE$ and $\angle EPA$ correspond to known arcs or can be deduced. Due to limited info, assume or conclude the measure of minor arc $\stackrel{\large{\frown}}{DE} = 2w - 3$ degrees. Final answer: The minor arc $\stackrel{\large{\frown}}{DE}$ measures $$2w - 3$$ degrees.