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To solve the problem, we need to find the length of \( \overline{MN} \) which is tangent to the given circle. Here's how to approach the problem step-by-step:

1. Given: The radius of the circle is 16, and the points \( M \) and \( N \) on the tangents are \( x - 3 \) and \( x + 3 \) respectively.

2. Let's denote the center of the circle by \( O \).

3. Since \( \overline{MN} \) is tangent to the circle at point \( L \), \( \overline{OL} \) is perpendicular to \( \overline{MN} \).

4. The right triangle \( \triangle OLN \) has:
   - \( \overline{OL} = 16 \) (radius)
   - \( \overline{LN} = x + 3 \) (part of the tangent line \( \overline{MN} \))

5. Using the Pythagorean theorem in \( \triangle OLN \):
   \[
   \overline{ON}^2 = \overline{OL}^2 + \overline{LN}^2
   \]
   Since \( \overline{ON} \) is the hypotenuse of the right triangle \( \triangle OLN \):
   \[
   \overline{ON}^2 = 16^2 + (x + 3)^2
   \]

6. Similarly, for the triangle \( \triangle OLM \), \( \overline{OM}^2 = 16^2 + (x - 3)^2 \).

However, we can simplify it by noting that the length of the tangent from an external point to a circle is the same, regardless of the point of tangency. Therefore:

7. Using the right triangle properties directly:
   \[
   MN = 2 \sqrt{x^2 - 16^2}
   \]
   This simplification follows by noting the same logic as steps 4-6 but effectively utilizing both lengths in the general tangent-segment theorem.

Thus:
\[
\overline{MN} = 2 \sqrt{x^2 - 256}
\]
Let's solve this expression for \( x \).

The length \( x \) can be found using the distance properties or specific analysis of \( \overline{MN} = 2(x-16) \) or geometric combinations confirming tangents from distances:

Here, \( \overline{MN} = \sqrt{(x+3) + (x-3)}\)

So \( MN = \boxed{\sqrt{(x+3)^2 + (x-3)^2} } = MN\)

Please confirm and details so to directly calculate \(x\) depending lengths. 

**Do you want further details or have any questions?**

Here are some related questions:

1. How do you find the length of a tangent from a point to a circle?
2. Can you explain the tangent-segment theorem in your own words?
3. What is the Pythagorean theorem and how is it applied in geometry?
4. How would you determine the coordinates of the points of tangency given the circle's equation?
5. Can you find the area of the triangle formed by the center of the circle and the points \( M \) and \( N \)?
6. How can you use coordinate geometry to solve problems involving circles and tangents?
7. What is the equation of a circle in standard form, and how can it be used to solve geometric problems?
8. Can you explain why the tangent to a circle is perpendicular to the radius at the point of tangency?

**Tip:** Always remember that the tangent to a circle from an external point is equal to each other regardless of tangency's distances.

Learn how to calculate the length of a tangent to a circle using the tangent-segment theorem and the Pythagorean theorem. This problem involves finding the distance of a tangent from an external point to a circle, with step-by-step calculations and geometric principles. Suitable for high school students studying geometry and circle theorems.

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