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The problem involves a triangle \( ABC \) with an incircle that touches \( AB \), \( BC \), and \( AC \) at points \( P \), \( Q \), and an unlabeled point on \( AC \), respectively. The side lengths are given as:

- \( AB = 6 \, \text{cm} \)
- \( BC = 7 \, \text{cm} \)
- \( AC = 8 \, \text{cm} \)

The question asks for the perimeter of \( \triangle APQ \).

### Solution Outline

The key to solving this problem lies in understanding the properties of a triangle's incircle and how it relates to the side lengths and segments created by points of tangency.

1. **Perimeter of Triangle \( ABC \)**:
   \[
   \text{Perimeter of } \triangle ABC = AB + BC + AC = 6 + 7 + 8 = 21 \, \text{cm}
   \]

2. **Segments on Each Side Due to the Incircle**:
   For a triangle with an incircle, the points of tangency divide each side into segments such that:
   - \( AP = s - BC \)
   - \( BP = s - AC \)
   - \( CQ = s - AB \)
   
   Here, \( s \) is the semi-perimeter:
   \[
   s = \frac{\text{Perimeter of } \triangle ABC}{2} = \frac{21}{2} = 10.5 \, \text{cm}
   \]

   Applying this:
   - \( AP = s - BC = 10.5 - 7 = 3.5 \, \text{cm} \)
   - \( BP = s - AC = 10.5 - 8 = 2.5 \, \text{cm} \)
   - \( CQ = s - AB = 10.5 - 6 = 4.5 \, \text{cm} \)

3. **Perimeter of \( \triangle APQ \)**:
   The perimeter of \( \triangle APQ \) is the sum of the sides \( AP \), \( PQ \), and \( QA \). Since \( PQ \) and \( QA \) correspond to segments created by the incircle's tangency:
   \[
   \text{Perimeter of } \triangle APQ = AP + PQ + QA
   = 3.5 + 4.5 + 2.5 = 10.5 \, \text{cm}
   \]

Thus, the perimeter of \( \triangle APQ \) is **10.5 cm**.

Would you like further details on the steps? 

### Related Questions

1. What is the formula for the semi-perimeter of a triangle, and how is it useful?
2. How does the incircle of a triangle divide each side into segments?
3. How can the properties of tangency be used to find segment lengths in other geometric shapes?
4. How would the solution change if one of the side lengths were altered?
5. What is the significance of the incircle's radius in determining triangle properties?

### Tip
The concept of semi-perimeter is extremely useful in geometry, especially when dealing with triangles that have an incircle.

Given a triangle ABC, where AB = 6 cm, BC = 7 cm, and AC = 8 cm. An incircle is inscribed in the triangle, touching sides AB, BC, and AC at points P, Q, and an unlabeled point, respectively. What is the perimeter of triangle APQ?

Learn how to determine the perimeter of triangle APQ, where an incircle is inscribed in triangle ABC with sides AB = 6 cm, BC = 7 cm, and AC = 8 cm. This problem covers essential geometric concepts, including the properties of tangency, segment lengths, and the use of the semi-perimeter.

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