To solve the problem, we need to determine the value of $x$ that makes $\overline{RQ}$ tangent to the circle at point $Q$.

Tangency Condition:

The tangency condition tells us that $\overline{RQ}$ is perpendicular to the radius $\overline{PQ}$ at point $Q$.

Using the Pythagorean Theorem:

Since $\overline{RQ}$ is tangent and perpendicular at $Q$, we know the right triangle formed is $\triangle PRQ$. The Pythagorean Theorem applies: $PR^2 = PQ^2 + RQ^2$

Where:

• $PR = x$
• $PQ = 9$ (radius of the circle)
• $RQ = 12$

Substituting the known values:

$x^2 = 9^2 + 12^2$

Simplifying:

$x^2 = 81 + 144$ $x^2 = 225$

Solving for $x$:

$x = \sqrt{225}$ $x = 15$

Final Answer:

The value of $x$ that makes $\overline{RQ}$ tangent to the circle is: $\boxed{15}$

Would you like me to explain the concept of tangents further or provide more examples?

Here are 5 related questions to deepen your understanding:

1. How does the Pythagorean Theorem apply to other tangent problems involving circles?
2. What is the geometric significance of a tangent line being perpendicular to the radius at the point of tangency?
3. Can you derive a general formula for tangents from external points to a circle?
4. What happens if the circle is not centered at the origin? How do we calculate tangents in such cases?
5. How can this principle be extended to three-dimensional geometry with spheres?

Tip: Always check if a triangle formed in a geometry problem is right-angled; this can simplify calculations significantly!