The shortest altitude corresponds to the longest side, which is 15 cm. Using area formula $ A = \frac{1}{2} \times \text{base} \times \text{height} $:

["### The Shortest Altitude Corresponds to the Longest Side at 15 cm: A Deep Dive Using Area Formula", "When exploring triangle geometry, one fundamental principle repeatedly surfaces: the shortest altitude corresponds to the longest side. This relationship becomes especially intuitive when we analyze the area formula of a triangle:\n[ A = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} ]", "In this article, we’ll explore why the shortest altitude in a triangle is directly linked to the triangle’s longest side—and how this geometric truth unfolds through the strategic use of area calculations.", "---", "### Understanding Triangle Altitudes and Sides", "In any triangle, the altitude (or height) is the perpendicular distance from a vertex to the opposite side (the base). While all three altitudes support the triangle’s area, their lengths vary depending on the length of the base they correspond to.", "Key insight:\nFor a fixed area, the altitude is inversely proportional to the base length.\nThis means longer sides have shorter corresponding altitudes—and vice versa.", "Given that, if the longest side measures 15 cm, its corresponding altitude must be the shortest among the three altitudes of the triangle.", "---", "### The Area Formula as Enlightening Tool", "Let’s apply the standard area formula to unlock clarity:", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Suppose the triangle has sides ( a ), ( b ), and ( c ), with ( c = 15 ) cm being the longest side. The corresponding altitude ( h_c ) satisfies:", "[\nA = \frac{1}{2} \ imes 15 \ imes h_c\n]", "Because ( c ) is the longest side, substituting it into the formula means:\n- If ( c > a ) and ( c > b ), then for the same area ( A ), ( h_c ) must be smaller than ( h_a ) and ( h_b ) — confirming ( h_c ) is the shortest altitude.", "---", "### A Numerical Example Illustrating the Principle", "Let’s assume a specific triangle for clarity. Consider a triangle with sides:", "- ( a = 10 ) cm\n- ( b = 12 ) cm\n- ( c = 15 ) cm\n(Note: This satisfies the triangle inequality ( a + b > c ), ( a + c > b ), ( b + c > a ), making it a valid triangle.)", "Using Heron’s formula for area:", "- Semi-perimeter: ( s = \frac{10 + 12 + 15}{2} = 18.5 ) cm\n- Area:\n[\nA = \sqrt{18.5(18.5 - 10)(18.5 - 12)(18.5 - 15)} = \sqrt{18.5 \ imes 8.5 \ imes 6.5 \ imes 3.5}\n\approx \sqrt{3600} = 60 \ ext{ cm}^2\n]", "Now compute altitudes:", "- ( h_a = \frac{2A}{10} = \frac{120}{10} = 12 ) cm\n- ( h_b = \frac{2A}{12} = \frac{120}{12} = 10 ) cm\n- ( h_c = \frac{2A}{15} = \frac{120}{15} = 8 ) cm", "Indeed, the shortest altitude ( h_c = 8 ) cm corresponds exactly to the longest side ( c = 15 ) cm, validating the geometric principle.", "---", "### Practical Implications", "Understanding this relationship helps in various mathematical and real-world applications:", "- Problem-solving: When given triangle sides and the area, identifying the longest side and computing its altitude quickly confirms it’s the shortest height.\n- Design & Engineering: In architecture and construction, knowing that larger structural sides require smaller supporting altitudes helps optimize designs and ensure stability.\n- Education: This concept strengthens foundational understanding of triangle relationships and reinforces algebraic interpretation of area.", "---", "### Summary", "- The shortest altitude in a triangle corresponds directly to the longest side.\n- This follows naturally from the area formula ( A = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} );\n- For a given area, longer bases yield shorter altitudes, and shorter bases yield longer altitudes.\n- Real-world examples and calculations demonstrate this principle powerfully.", "Whether you're solving geometry problems or analyzing structural designs, recognizing that the shortest altitude matches the longest side enhances precision and insight—turning abstract formulas into tangible understanding.", "---", "Keywords: altitude in triangle, triangle area formula, shortest altitude corresponds to longest side, triangle geometry, Heron’s formula, base-height formula, geometry principles, mathematical application, triangle side relationships."]