The altitude corresponding to a side is $ h = \frac{2A}{\text{side}} $. The longest altitude corresponds to the shortest side, which is 13 cm:
["Understanding Altitude and Side Length in Triangles: The Formula $ h = \frac{2A}{\ ext{side}} $", "In geometry, the relationship between a triangle’s area, altitude, and side lengths is fundamental to understanding triangle properties. One key formula frequently used is:", "> $ h = \dfrac{2A}{\ ext{side}} $", "This equation defines the altitude $ h $ drawn from a vertex opposite a given side, where $ A $ is the total area of the triangle. Since altitude is inversely proportional to the side it drops on, the longest altitude corresponds to the shortest side—this principle holds true for any triangle.", "In particular, when the shortest side measures just 13 cm, the corresponding altitude is maximized. Using the formula, if we denote $ s_{\min} = 13 , \ ext{cm} $, the altitude $ h $ is:", "$$\nh = \frac{2A}{13}\n$$", "This clearly shows that without knowing area $ A $, the height for the shortest side scales directly. Maximizing $ h $ means minimizing the base, confirming the logic that the shortest side yields the greatest altitude.", "---", "Why the Shortest Side Determines the Longest Altitude", "Consider a triangle with three different side lengths: one side is very short (e.g., 13 cm), while the others are longer. Because the altitude is defined as twice the area divided by the base length, smaller bases force the altitude larger—assuming a fixed area.", "Thus, among all altitudes, the one drawn to the shortest side is always the longest. This principle applies universally across all triangle types—acute, obtuse, equilateral, or scalene.", "---", "Applying the Formula $ h = \frac{2A}{s} $ to Real-World Problems", "Whether calculating structural loads, designing trusses, or solving geometric puzzles, knowing how altitude depends on side length is invaluable. Suppose a triangle has an area of $ A = 39, \ ext{cm}^2 $. Then, when the side length is $ s = 13, \ ext{cm} $, the altitude is:", "$$\nh = \frac{2 \ imes 39}{13} = 6 , \ ext{cm}\n$$", "This calculation becomes a quick way to estimate heights when dimensions are mixed, aiding in design and analysis.", "---", "Key Takeaways", "- The altitude of a triangle is inversely proportional to the length of the side it bisects.\n- The longest altitude always corresponds to the shortest side.\n- With $ s = 13, \ ext{cm} $, altitude $ h $ is maximized, formally: $ h = \dfrac{2A}{13} $, where $ A $ is the area.\n- Use this relationship for geometry problem-solving, engineering calculations, and architectural design.", "---", "Conclusion", "Understanding how altitude relates to side lengths through the formula $ h = \frac{2A}{s} $ enables clearer insight into triangle behavior. The shortest side produces the longest altitude, a foundational principle anyone studying geometry, design, or physics should grasp. Next time you analyze a triangle, remember: measuring the base thinks you’ll unlock the height."]
