In this image, we are working with a triangle with an inscribed circle, and the goal is to find the length of segment AD. To fill in the blanks, let's go step by step according to the tangent-segment theorem, which states that the tangents drawn from an external point to a circle are equal in length.

Step-by-Step Fill-in:

1. AD = AF = x (Tangent Segment Theorem)
   According to the tangent-segment theorem, the tangents drawn from point A to the circle are equal. Thus, AD = AF = x.

2. AC = AF + CF (A-F-C)
   This follows from the fact that AC is the total length of AF and CF.

3. 10 = x + CF
   Since AC = 10 cm, and AC = AF + CF, we substitute:
   10 = x + CF.

4. CF = CE (Tangent Segment Theorem)
   By the tangent-segment theorem, CF = CE, because the tangents from point C to the circle are equal.

5. CE = BE (Tangent Segment Theorem)
   Similarly, CE = BE because BE is the tangent drawn from point B.

6. CE + BE = CB (C-E-B)
   The sum of CE and BE is the length of side CB, as they lie along the same line segment.

7. BE = 8
   From the given data, we know that BC = 8 cm. So, BE = 8 cm.

8. BD = BE (Tangent Segment Theorem)
   From the tangent-segment theorem, BD = BE, as BD is the tangent from point B.

9. BD = x - 2
   From the equation of AD, we know that BD = x - 2.

10. AD + BD = AB (A-D-B)
    AD + BD = AB, which is the total length of side AB (12 cm).

11. x + (x - 2) = 12
    This results from substituting the expressions for AD and BD. Simplifying, we get: 2x - 2 = 12.

12. 2x = 14
    Solving the equation, we get: x = 7.

13. AD = 7 cm
    Therefore, the length of AD is 7 cm.

Final Answer:

AD = 7 cm.

Let me know if you'd like further details on any of these steps! Here are some related questions you can explore:

1. How does the tangent-segment theorem apply to other geometric shapes like squares or rectangles?
2. What is the relationship between the semiperimeter of a triangle and the sides in this context?
3. How can we apply this method to find the inradius of a triangle?
4. Can you explain how the incenter of a triangle relates to the inscribed circle?
5. What other geometric properties can help us solve for missing sides of triangles with incircles?

Tip: Always look for equal tangents when solving problems involving incircles in triangles!