The problem shows a circle with a central angle $\angle AOC = (3x - 12)^\circ$ and an inscribed angle $\angle ABC = 60^\circ$. The goal is to find the value of $x$.

Key concepts:

1. Central Angle and Inscribed Angle Relationship: The inscribed angle subtended by the same arc is half the measure of the central angle subtended by that arc. In this case, $\angle ABC = \frac{1}{2} \times \angle AOC$.

Steps to solve:

• $\angle ABC = 60^\circ$
• According to the inscribed angle theorem, $\angle ABC = \frac{1}{2} \times \angle AOC$
• Substitute the values: $60 = \frac{1}{2} \times (3x - 12)$
• Multiply both sides by 2 to eliminate the fraction: $120 = 3x - 12$
• Add 12 to both sides: $132 = 3x$
• Divide by 3: $x = 44$

Answer:

The value of $x$ is 44, so the correct option is E.

Would you like more details or explanations on this? Here are some related questions:

1. What is the difference between a central angle and an inscribed angle in a circle?
2. How does the inscribed angle theorem help in solving problems like this?
3. Can the inscribed angle ever be larger than the central angle subtending the same arc?
4. How can we prove the inscribed angle theorem?
5. How does this problem change if the angles were expressed in radians instead of degrees?

Tip: Always remember that the inscribed angle is half the measure of the central angle subtended by the same arc.