Given:
- Triangle ABC is a right-angled triangle.
- Triangle ABC circumscribes a circle of radius R.
- AB and BC are of length 8 cm and 6 cm respectively.

To Find:
The radius of the inscribed circle, R.

Solution:
To find the radius of the inscribed circle, we can make use of the properties of a right-angled triangle and the relationship between the radius of the inscribed circle and the sides of the triangle.

Step 1: Identify the Properties of a Right-Angled Triangle
A right-angled triangle has the following properties:
1. One angle measures 90 degrees (the right angle).
2. The sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side (Pythagorean theorem).

Step 2: Use the Pythagorean Theorem
In triangle ABC, the side lengths are given as AB = 8 cm and BC = 6 cm. Let AC be the longest side (hypotenuse).

Using the Pythagorean theorem, we can find the length of AC:
AC^2 = AB^2 + BC^2
AC^2 = 8^2 + 6^2
AC^2 = 64 + 36
AC^2 = 100
AC = 10 cm

Step 3: Identify the Relationship between the Radius and Sides of the Triangle
In a right-angled triangle, the radius of the inscribed circle is given by the formula:

R = (AB + BC - AC) / 2

Step 4: Substitute the Given Values to Find R
R = (8 + 6 - 10) / 2
R = 4 / 2
R = 2 cm

Step 5: Conclusion
The radius of the inscribed circle, R, is 2 cm.

Summary:
- Triangle ABC is a right-angled triangle with AB = 8 cm, BC = 6 cm, and AC = 10 cm.
- The radius of the inscribed circle, R, is found using the formula R = (AB + BC - AC) / 2.
- Substituting the values, we find that R = 2 cm.