Solution:

Given:
- Tangents AP and AQ are drawn from an external point P to a circle with center O.
- Tangents AP and AQ are perpendicular to each other.
- Length of each tangent is 5 cm.

To find: The radius of the circle.

Let's solve this problem step by step:

Step 1: Understanding the problem

- We have a circle with center O.
- Tangents AP and AQ are drawn from an external point P to the circle.
- Tangents AP and AQ are perpendicular to each other.
- The length of each tangent is 5 cm.

Step 2: Identifying the important information

- The length of each tangent is 5 cm.
- The tangents are perpendicular to each other.
- We need to find the radius of the circle.

Step 3: Using the given information

- We know that tangents drawn from an external point to a circle are equal in length.
- Therefore, AP = AQ = 5 cm.
- We also know that if two tangents are drawn from an external point to a circle, the line joining the external point to the center of the circle bisects the angle between the tangents.
- In this case, PO bisects the angle between AP and AQ.

Step 4: Applying the properties of perpendicular lines

- Since AP and AQ are perpendicular to each other, the angle between them is 90 degrees.
- Therefore, the angle between AP and PO is 45 degrees (as PO bisects the angle between AP and AQ).
- Similarly, the angle between AQ and PO is also 45 degrees.

Step 5: Applying trigonometry

- In a right-angled triangle, the tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
- In triangle APO, the angle APO is 45 degrees, and the side opposite this angle is AP (which is 5 cm).
- So, the tangent of angle APO is equal to AP/OP.
- Similarly, in triangle AQO, the tangent of angle AQO is equal to AQ/OP.
- Since the tangents of both angles are equal, we have AP/OP = AQ/OP.
- Simplifying this equation, we get AP = AQ.

Step 6: Calculating the radius

- Since AP = AQ = 5 cm, we can conclude that the distance from the center of the circle to the point of contact of the tangent is 5 cm.
- But the radius of the circle is the distance from the center to any point on the circumference.
- Therefore, the radius of the circle is 5 cm.

Hence, the correct answer is (c) 5 cm.