1. Problem Statement: Given a triangle SLM with sides SL=13m, LM divided into segments 2m and 5m, and SM labeled as $k$, find the length of $k$ using the given information and relationships. 2. Given equation: $$3H^2 = 3L^2 + LM^2$$ 3. Substituting: $$3H^2 = 13^2 + 5^2$$ 4. Calculate squares: $$169 + 25 = 194$$ 5. Simplify further: $$x^2 = 194$$ 6. Taking square root of both sides: $$x = \sqrt{194}$$ 7. Numerical evaluation: $$x \approx 13.9284$$ 8. Interpretation: The length $k$ (SM) is approximately $13.93$ meters. 9. Note: The height $h$ and segments of $LM$ are given but not explicitly needed to find $k$ from the Pythagorean relation provided. Final answer: $$k \approx 13.93 m$$