To solve this problem, let’s analyze the properties of tangent and secant segments.

Problem Breakdown

• Given: Segment $PQ$ is tangent to the circle at point $Q$, and $PR$ is a secant line that intersects the circle at points $R$ and $S$.
• Goal: Identify the equation that relates the lengths of the tangent segment $PQ$ and the segments $PR$ and $PS$ on the secant line.

The Tangent-Secant Theorem

The Tangent-Secant Theorem states: $(PQ)^2 = (PR) \times (PS)$ This equation describes the relationship between a tangent and a secant segment drawn from the same external point $P$. Here:

• $PQ$ is the length of the tangent.
• $PR$ is the entire length of the secant segment from $P$ to $R$.
• $PS$ is the entire length of the secant segment from $P$ through $R$ to $S$.

Applying the Tangent-Secant Theorem to the Options

Based on the theorem, the correct answer is: $\text{Option A: } (PQ)^2 = (PR) \times (PS)$

Final Answer

The correct choice is A.

Would you like further explanation on this theorem or have any other questions?

Here are some related questions you might find helpful:

1. What are other important circle theorems related to tangents and secants?
2. How does the Tangent-Secant Theorem apply to different configurations?
3. What is the difference between a tangent and a secant line in a circle?
4. Can this theorem be used with inscribed angles in the circle?
5. How do you prove the Tangent-Secant Theorem?

Tip: Remember that tangent and secant line relationships often appear in geometry problems involving circles, so knowing the theorems can help you solve such problems quickly.