Question: A science communicator explains geometric scaling using a triangle with side lengths $ 3 $, $ 4 $, and $ 5 $. If each side is scaled by $ k $, and the inradius of the scaled triangle is $ 6 $, find the value of $ k $.
["A science communicator explains geometric scaling using a triangle with side lengths $ 3 $, $ 4 $, and $ 5 $. If each side is scaled by $ k $, and the inradius of the scaled triangle is $ 6 $, find the value of $ k $.", "Why not every geometry lesson captures curiosity quite like a right triangle’s elegant proportions? The classic $ 3-4-5 triangle remains a foundation in understanding geometric scaling—simple, recognizable, and deeply connected to mathematical relationships. Today, a growing audience is exploring how scaling such a triangle affects intrinsic properties, including its inradius. This question isn’t just academic: scaling a triangle alters its size, but key characteristics like area, perimeter, and notably its inradius—offering insight into how geometry governs shape behavior under transformation. Understanding these shifts helps in fields ranging from architecture to data visualization, where proportional relationships matter.", "---", "### The Science Behind Inradius and Triangle Scaling", "In geometry, the inradius of a triangle—its radius of the inscribed circle—is a crucial measure linking shape dimensions to internal space. For a triangle, the inradius $ r $ relates to area $ A $ and semi-perimeter $ s $ via the formula: \n$$ r = \frac{A}{s} $$ \nWhen a triangle is scaled uniformly by a factor $ k $, all linear dimensions multiply by $ k $. This scales the area by $ k^2 $ and the semi-perimeter by $ k $, meaning the inradius scales proportionally to $ k $. Specifically, if original $ r = 3 $, then scaled inradius becomes $ k \ imes 3 $. Given the scaled inradius is $ 6 $, solving \n$$ 3k = 6 $$ \nimmediately yields $ k = 2 $. This elegant proportionality reveals scaling’s predictable impact on triangle geometry.", "---", "### Scaling the $ 3-4-5 Tri"]
