To solve this, we need to determine the arc measure of $\overset{\frown}{BC}$.

Step 1: Identify key information from the diagram

• The three angles at the center of the circle $P$ are:
  • $\angle BPA = 20y - 11$ degrees,
  • $\angle APC = 7y - 7$ degrees,
  • $\angle BPC = 4y + 6$ degrees.

Step 2: Use the fact that the angles around a point sum up to 360°

Since all the angles are centered at $P$ and form a complete circle, we know:

$(20y - 11) + (7y - 7) + (4y + 6) = 360^\circ$

Step 3: Simplify and solve for $y$

$20y - 11 + 7y - 7 + 4y + 6 = 360$ Combine like terms: $(20y + 7y + 4y) + (-11 - 7 + 6) = 360$ $31y - 12 = 360$ Solve for $y$: $31y = 360 + 12$ $31y = 372$ $y = \frac{372}{31} = 12$

Step 4: Calculate the arc measure of $\overset{\frown}{BC}$

The measure of arc $\overset{\frown}{BC}$ corresponds to $\angle BPC$, which is given as $4y + 6$.

Substitute $y = 12$ into this expression: $4(12) + 6 = 48 + 6 = 54^\circ$

Thus, the arc measure of minor arc $\overset{\frown}{BC}$ is $54^\circ$.

Final Answer:

$\boxed{54^\circ}$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you find the arc length given the radius and angle of a circle?
2. What is the difference between a major and a minor arc?
3. How can the central angle theorem be applied to solve problems in circles?
4. How do you solve for an unknown angle when given all other angles around a circle?
5. What is the relationship between a circle's central angle and its corresponding arc?

Tip: Always remember that the sum of angles around a point is $360^\circ$, which is crucial for solving circle-based angle problems.