To solve this problem, let's first analyze the given information and the figure:

1. Description of the Problem:

   • $RT$ is a tangent to circle $R$ at point $T$.
   • $\overline{RT} = 6$.
   • $\overline{RC} = 4$.
   • We need to find the ratio $\frac{TR}{RC}$.

2. Understanding the Geometry:

   • $RT$ being a tangent means it touches the circle at exactly one point, $T$, which is on the circle.
   • $R$ is the center of the circle.
   • $RC$ is a radius of the circle.

3. Relevant Theorem:

   • The Power of a Point Theorem (also known as the Tangent-Secant Theorem) states that if a tangent from an external point intersects a secant that passes through the circle, the square of the length of the tangent segment equals the product of the lengths of the external and internal segments of the secant.

   In our scenario:

   • $RT^2 = RC \cdot RB$, where $RB$ is the length of the secant line extending from $R$ to $B$.

4. Application to the Problem:

   • Using the information $RT = 6$ and $RC = 4$:

   $RT^2 = RC \cdot RB$

   $6^2 = 4 \cdot RB$

   $36 = 4 \cdot RB$

   $RB = \frac{36}{4} = 9$

5. Find $TR$:

   • Since $RT$ is a tangent and $RC$ is a radius intersecting the tangent at $T$, $RT$ is perpendicular to $RC$ at $T$. Hence, we are dealing with a right-angled triangle $RTC$ where:

   $RT = 6 \quad \text{and} \quad RC = 4.$

   We use the Pythagorean theorem to find $TR$:

   $TR^2 = RT^2 + RC^2$

   $TR^2 = 6^2 + 4^2$

   $TR^2 = 36 + 16$

   $TR^2 = 52$

   $TR = \sqrt{52} = 2\sqrt{13}$

6. Find the Ratio $\frac{TR}{RC}$:

   $\frac{TR}{RC} = \frac{2\sqrt{13}}{4} = \frac{\sqrt{13}}{2}$

Therefore, the ratio $\frac{TR}{RC}$ is $\frac{\sqrt{13}}{2}$.

Would you like more details on any part of this solution, or do you have any specific questions?

Related Questions:

1. What is the Power of a Point theorem, and how is it derived?
2. How does the Pythagorean theorem apply to circles and tangents?
3. How do you prove that a tangent to a circle is perpendicular to the radius at the point of tangency?
4. What are the properties of tangents to a circle from a common external point?
5. How can we use geometric means to find the lengths of tangents from an external point to multiple circles?

Tip:

Always remember to verify if a given segment is a tangent to a circle. If it is, it will be perpendicular to the radius drawn to the point of tangency.