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The problem asks for the perimeter of triangle \( \triangle ABC \), which is circumscribed around circle \( D \). Given:

- \( AQ = 13 \, \text{cm} \)
- \( BR = 10.5 \, \text{cm} \)
- \( SC = 6 \, \text{cm} \)

By a property of a triangle circumscribed around a circle, the tangents from a common external point to the circle are equal in length. Thus, we have the following pairs of equal segments:
- \( AQ = AS \)
- \( BR = BQ \)
- \( CS = CR \)

Let's denote these equal segments:
- \( AS = AQ = 13 \, \text{cm} \)
- \( BQ = BR = 10.5 \, \text{cm} \)
- \( CR = CS = 6 \, \text{cm} \)

Now, to find the perimeter of triangle \( ABC \), we add the sides:
\[
\text{Perimeter} = AB + BC + CA
\]
Where:
\[
AB = AQ + BQ = 13 + 10.5 = 23.5 \, \text{cm}
\]
\[
BC = BR + CR = 10.5 + 6 = 16.5 \, \text{cm}
\]
\[
CA = CS + AS = 6 + 13 = 19 \, \text{cm}
\]

Thus, the perimeter of triangle \( ABC \) is:
\[
\text{Perimeter} = 23.5 + 16.5 + 19 = 59 \, \text{cm}
\]

The perimeter of triangle \( ABC \) is **59 cm**.

Would you like more details or further clarifications? Here are some related questions you might find interesting:

1. What is the formula for the area of a triangle circumscribed around a circle?
2. How do we derive the tangents' equal length property in such triangles?
3. How do we find the radius of the incircle given the sides of a triangle?
4. What are other properties of tangents drawn from an external point to a circle?
5. How can we find the area of triangle \( ABC \) if we know its perimeter and inradius?

**Tip:** In any triangle circumscribed around a circle, the tangents drawn from a common external point to the circle are always equal.

Triangle ABC is circumscribed around circle D. Segments AQ, BR, and SC measure 13, 10.5, and 6 cm, respectively. The perimeter of triangle ABC is:

Learn how to calculate the perimeter of triangle ABC, which is circumscribed around circle D. Given AQ = 13 cm, BR = 10.5 cm, and SC = 6 cm, discover the key geometric properties of tangents and use the perimeter formula. Suitable for students in Grades 9-10.

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