To solve this problem, we can apply the Power of a Point Theorem. Specifically, we'll use the tangent-secant theorem, which states:

$PA^2 = PB \times PC$

Given:

• $PB = 4$
• $BC = 8$
• Therefore, $PC = PB + BC = 4 + 8 = 12$

Applying the Theorem:

$PA^2 = PB \times PC = 4 \times 12 = 48$

Taking the square root of both sides: $PA = \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

Answer:

The length of $PA$ is $4\sqrt{3}$, which corresponds to option C.

Would you like more details or further clarification? Here are some related questions you might find useful:

1. What is the Power of a Point Theorem and how does it apply to circles?
2. How do you derive the tangent-secant theorem?
3. Can the Power of a Point Theorem be applied to other types of geometry problems?
4. What are some examples of secant-tangent relationships in real-world applications?
5. How does the relationship between the radius and tangent line of a circle help in solving geometric problems?

Tip: When dealing with circle geometry, always remember to explore the Power of a Point Theorem for secant and tangent combinations—it can save time and provide direct answers!