The shortest altitude corresponds to the longest side, which is \(c = 15\):
["# The Shortest Altitude Corresponds to the Longest Side: Proving with (c = 15)", "In geometry, understanding the relationship between triangle sides and altitudes is fundamental to solving problems in triangle properties, area computations, and optimization. One key principle is: the shortest altitude corresponds to the longest side. This concept becomes especially clear when analyzing a specific triangle where side (c = 15) is the longest. Let’s explore this relationship in depth.", "---", "## Understanding Altitudes and Sides in a Triangle", "An altitude of a triangle is a perpendicular line segment from a vertex to the line containing the opposite side (or its extension). Since area is constant regardless of which side is used as the base, each side has a uniquely defined altitude:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} \implies h = \frac{2 \ imes \ ext{Area}}{\ ext{base}}\n]", "This formula shows that altitude (h) is inversely proportional to the length of the base segment. Thus, the longer the side (base), the shorter the corresponding altitude — assuming fixed area.", "---", "## The Case Where (c = 15) Is the Longest Side", "Consider a triangle with sides (a), (b), and (c), where (c = 15) is the longest side. Because of the inverse relationship:", "- Side (c) is the longest base\n- Therefore, altitude (h_c) drawn to side (c) will be the shortest among the three altitudes", "This geometric property holds universally for any triangle with a longest side of length 15. Whether (a) and (b) are shorter or longer (as long as (c) remains the largest), (h_c) remains the shortest altitude.", "---", "## How to Prove This Using the Area Formula", "Let the area of the triangle be (A). Then:", "[\nh_c = \frac{2A}{c} = \frac{2A}{15}\n]", "No matter the shape or angles of the triangle, (A) is fixed for a given triangle. Since 15 is largest, (h_c = \frac{2A}{15}) must be smaller than (h_a = \frac{2A}{a}) or (h_b = \frac{2A}{b}), where (a < 15), (b < 15).", "Hence, altitude relative to side (c) is minimized — confirming (h_c) is the shortest.", "---", "## Practical Implications in Problem Solving", "This principle is instantly useful when:\n- Calculating altitudes given area and longest side\n- Comparing altitudes across different triangles\n- Setting up equations involving triangle area and perimeter constraints", "For example, if you know a triangle has sides 9, 15, and 17, and wish to find shortest altitude, identifying (c = 17) as longest side tells you immediately that its corresponding (h_c = \frac{2A}{17}) is shortest — saving time in solving for area or other altitudes.", "---", "## Summary", "| Property | Explanation |\n|-------------------------|------------------------------------------------------------|\n| Longest side (c = 15) | Largest base ( \Rightarrow ) shortest altitude (h_c) |\n| Inverse relationship | ( \ ext{altitude} \propto \frac{1}{\ ext{side length}} ) |\n| Area formula support | ( h_c = \frac{2A}{15} ) is minimized |\n| Applications | Solving for unknown altitudes, triangle area estimation |", "---", "Conclusion:\nThe shortest altitude in any triangle always opposes the longest side. For triangles where (c = 15), this means the altitude to side (c) is the shortest possible, directly tied to the triangle’s area and side lengths. Mastering this relationship strengthens your ability to analyze and solve geometry problems efficiently.", "Whether you're studying triangles for exams or tackling advanced math challenges, recognizing how altitude and side length balance reveals deeper insight into triangle geometry.", "---", "Keywords for SEO optimization:\naltitude and side length relationship, shortest altitude corresponds to longest side, triangle altitude formula, geometry principle short altitude, longest side height shortest formula, triangle area altitude proof, why shortest altitude corresponds to longest side."]
