Problem Statement:
Two concentric circles have an equal center O, and the chord AN of the bigger circle touches the smaller circle at point P. Given that OP = 3 cm and AN = 8 cm, we need to find the radius of the bigger circle.

Explanation:
To solve this problem, we can use the concept of tangents and chords.

1. Basic Definitions:
Let's define some basic terms:
- O: The center of both circles.
- R: The radius of the bigger circle.
- r: The radius of the smaller circle.
- P: The point where the chord AN touches the smaller circle.

2. Tangents and Chords:
When a chord touches a circle at a specific point, it is always perpendicular to the radius drawn to that point. Therefore, we can draw a perpendicular line from point P to the chord AN.

3. Drawing the Diagram:
To visualize the problem, draw two concentric circles with center O. Mark the point P where the chord AN touches the smaller circle. Draw the radius OP and the chord AN.

4. Applying the Tangent-Chord Rule:
According to the tangent-chord rule, the length of the line segment from the center O to the point of contact P is equal to the geometric mean of the lengths of the two segments into which the chord AN is divided.

In this case, we have:
OP * OP = AP * PN

Substituting the given values, we get:
3 * 3 = AP * PN

Simplifying further, we have:
AP * PN = 9

5. Solving for AP and PN:
To find the lengths of AP and PN, we can use the Pythagorean theorem. Since OP = 3 cm and AP = r (radius of the smaller circle), we have:
r * r + 3 * 3 = r * r + PN * PN

Simplifying the equation, we get:
9 = PN * PN

Therefore, PN = 3 cm.

Since AP * PN = 9, we can find AP by dividing 9 by PN:
AP = 9 / 3 = 3 cm.

6. Finding the Radius of the Bigger Circle:
In the larger circle, the chord AN is also a diameter. Therefore, the radius of the bigger circle is half the length of the chord AN.

Given that AN = 8 cm, the radius of the bigger circle is:
R = AN / 2 = 8 / 2 = 4 cm.

Conclusion:
The radius of the bigger circle is 4 cm.