The area is 84 m². The shortest altitude corresponds to the longest side, which is 15 m. Using $A = \frac{1}{2} \times \text{base} \times \text{height}$:

["# Understanding Area Using Altitude and Base: A Case Study", "A key concept in geometry is how area depends on the base and its corresponding altitude. In this article, we explore a specific geometric figure where the area is 84 m², and analyze how the shortest altitude relates to the longest side—highlighting the powerful formula $ A = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} $.", "## The Area Formula…", "The formula for the area of a triangle is simple yet vital:", "$$\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n$$", "Here, $ A $ represents the area, the base is the length of one side of the triangle, and the height (or altitude) is the perpendicular distance from that base to the opposite vertex.", "## The Triangle in Focus", "We’re examining a triangle with a total area of 84 m². Among its three sides, one is the longest, measuring 15 meters. Since area depends on both the base and the corresponding height, the shortest altitude must correspond to the longest base—this naturally minimizes the required height, keeping area constant.", "Let’s assign:\n- Base = 15 m (longest side)\n- Area, $ A = 84 $ m²\n- Height = $ h $ (perpendicular from the opposite vertex to the 15 m base)", "Using the area formula:", "$$\n84 = \frac{1}{2} \ imes 15 \ imes h\n$$", "Multiply both sides by 2:", "$$\n168 = 15 \ imes h\n$$", "Solve for $ h $:", "$$\nh = \frac{168}{15} = 11.2 \ ext{ m}\n$$", "So, the shortest altitude is 11.2 meters—corresponding to the longest side of 15 meters.", "## Why the Shortest Altitude Fits Here", "Because area is fixed at 84 m², increasing the base while reducing the height maintains a constant area. The longest side (15 m) needs a relatively short height to balance the formula. Hence, 11.2 m is the shortest possible altitude—any longer height paired with the 15 m base would overshoot the area.", "This relationship clarifies how bases and heights work in concert:\n- A longer base requires a shorter height for constant area.\n- The shortest altitude always matches the longest side.", "## Conclusion", "Whether calculating area for architectural design, land surveying, or classroom learning, understanding the inverse relationship between base and altitude is essential. With an area of 84 m² and a longest side of 15 m, the shortest altitude measures 11.2 m—proving how $ A = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} $ accurately models real-world geometry.", "If you're working with triangles or area calculations, remember: the tallest altitude corresponds to the shortest base, and vice versa—always keeping $ A = \frac{1}{2} \ imes b \ imes h $ in focus.", "---\nKeywords: triangle area formula, altitude and base relationship, area calculation 84 m², geometry problem solver, perpendicular height in triangles"]