The altitude corresponding to a side is $ h = \frac{2A}{\text{side}} $. The longest altitude corresponds to the shortest side, which is 13 m:

["Understanding the Altitude of a Triangle: How Side Lengths Determine the Longest Altitude", "When analyzing triangles, one key relationship is the correlation between a triangle’s side lengths and its altitudes. A fundamental principle is that the longest altitude corresponds to the shortest side, and this relationship is mathematically expressed as:", "[\nh = \frac{2A}{\ ext{side}}\n]", "where ( h ) is the length of the altitude to the given side, and ( A ) is the area of the triangle.", "### The Key Insight: Inverse Proportionality Between Altitudes and Side Lengths", "Because the area ( A = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} ), rearranging gives the altitude formula ( h = \frac{2A}{\ ext{side}} ). This equation reveals an essential geometric truth: the altitude is inversely proportional to the corresponding side length.", "This means:\n- Shorter sides produce taller altitudes.\n- Longer sides produce shorter altitudes.", "For any triangle, the shortest side must correspond to the longest altitude, and in many natural and engineered forms, the shortest side is often constrained or measured precisely—such as in structural supports or architectural elements.", "### Case Study: The Triangle with Shortest Side of 13 Meters", "Let’s consider a specific triangle where the shortest side measures exactly 13 meters. Applying the formula:", "[\nh = \frac{2A}{13}\n]", "Even without knowing the full area, this formula confirms that this is the altitude from the vertex opposite the 13-meter side, and it is the longest altitude in the triangle.", "If you’mapping this to real-world applications—like roof trusses, bridge supports, or triangle-based steel frameworks—knowing this relationship helps engineers optimize height and strength based on side lengths.", "### Why This Matters in Geometry and Design", "Understanding that the highest altitude belongs to the shortest side allows for smarter design decisions. For example:\n- Maximize structural clearance by orienting taller supports (longer altitudes) along narrower or shorter edges.\n- Calculate required reinforcement based on perpendicular heights, helping prevent failure at stress points.\n- Optimize space usage in triangular shapes while maintaining stability.", "### Summary", "The equation ( h = \frac{2A}{\ ext{side}} ) elegantly captures a core geometric truth: in any triangle, the altitude increases as the base decreases. In cases where the shortest side is 13 meters, the associated altitude is the triangle’s tallest possible height—vital knowledge in architecture, engineering, and triangular tessellation.", "---", "Key Takeaways:\n- Altitude ( h ) is inversely proportional to the triangle’s side length.\n- The shortest side has the longest altitude.\n- In a triangle with a 13-meter side, the corresponding altitude is ( h = \frac{2A}{13} ).\n- Use this relationship to design efficient and stable triangular structures.", "Understanding this principle empowers better analysis and application of triangular forms across science and engineering disciplines."]