Triangle ABC is circumscribed around circle D. Segments AQ, BR and SC measure…

Concept: A tangent theorem for circles inscribed in a triangle: the two tangent segments drawn from a single external point (a vertex of the triangle) to a circle are always equal in length. So when a circle is inscribed in a triangle, each vertex contributes one pair of equal tangent segments running along the two sides that meet at that vertex.

Application: In the figure, the incircle D touches side AB at Q, side BC at R, and side AC at S.

1. From vertex A, the two tangents are AQ and AS, so AS = AQ = 13 cm.

2. From vertex B, the two tangents are BQ and BR, so BQ = BR = 10.5 cm.

3. From vertex C, the two tangents are CR and CS, so CR = CS = 6 cm.

4. Side AB = AQ + QB = 13 + 10.5 = 23.5 cm.

5. Side BC = BR + RC = 10.5 + 6 = 16.5 cm.

6. Side CA = CS + SA = 6 + 13 = 19 cm.

7. Perimeter = AB + BC + CA = 23.5 + 16.5 + 19 = 59 cm.

Cross-check: The same total can be reached independently: each of the three given tangent lengths (13, 10.5, 6) belongs to exactly two sides of the triangle (once at each of its two endpoints’ tangent pairs), so the perimeter equals twice their sum: 2 × (13 + 10.5 + 6) = 2 × 29.5 = 59 cm, which matches the side-by-side total above and confirms the perimeter is 59 cm.