To find the longest altitude, we use the shortest side, since altitude is inversely proportional to the base. The shortest side is $ 7 $ cm.
["Understanding the Longest Altitude: Why the Shortest Side Dictates Maximum Height in Geometry", "When studying triangles, one fundamental principle guides the relationship between a triangle’s base and its altitude: altitude is inversely proportional to the base length, provided the area remains constant. This crucial insight reveals that the longest altitude corresponds directly to the shortest side of the triangle. In this article, we explore why this inverse relationship holds, why the shortest side (measured at $ 7 $ cm) is key to determining the triangle’s maximum altitude, and how this principle applies across geometry applications.", "### What Is Altitude and Why Is It Inversely Proportional to Base Length?", "Altitude, or height, in a triangle is defined as the perpendicular distance from a vertex to the opposite side (the base). The area ( A ) of a triangle is calculated using the formula:", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Rearranged, height (altitude) can be expressed as:", "[\n\ ext{height} = \frac{2A}{\ ext{base}}\n]", "Since the area ( A ) and the shape’s constant geometry determine a fixed value for these variables, height decreases as the base length increases—and vice versa. Therefore, the shortest side naturally produces the longest altitude because it maximizes height for a given base.", "### Why the Shortest Side Is Critical for Finding the Longest Altitude", "By definition, a triangle’s three sides vary in length: one is shortest, one is medium, and one is longest. When calculating altitudes, the side chosen as the base determines the associated altitude. Because the altitude depends inversely on the base length, using the shortest side ensures we maximize the height value—it gives the largest possible altitude for that triangle.", "### Mathematical Insight: The Formula Behind It", "Let’s denote:\n- ( a, b, c ) as the lengths of the triangle’s sides,\n- ( A ) as the constant area,\n- Altitude from vertex opposite side ( a ):\n[\nh_a = \frac{2A}{a}\n]", "Since ( a = 7 ) cm is the shortest side, plugging it into the formula yields the greatest height:", "[\nh_{\ ext{max}} = \frac{2A}{7}\n]", "Any larger base (side) would produce a smaller height, confirming the shortest side maximizes altitude.", "### Practical Example in Geometry and Real-World Applications", "Imagine a triangle with sides 5 cm, 7 cm, and 9 cm. The shortest side (5 cm) yields the longest altitude. Engineers, architects, and drafters rely on this principle when estimating maximum heights in triangular structures—such as roof designs or triangular supports—ensuring stability by maximizing vertical reach where feasible.", "### Summary", "- Altitude and base are inversely proportional in a triangle’s area formula.\n- The shortest side produces the longest altitude, since height = ( \frac{2A}{\ ext{base}} ).\n- Fixing the area and shape, minimizing the base maximizes height.\n- This principle is essential in geometry, physics, and design for structural analysis and optimization.", "### Key Takeaway", "To find the longest altitude in a triangle, identify and use the shortest side as the base. This inverse relationship ensures accurate and efficient identification of maximal height in triangular figures, supporting applications across science, engineering, and education.", "---", "Keywords: triangle altitude, inverse proportion altitude, shortest side longest altitude, triangle geometry, altitude formula, base-height relationship, geometric principle, structural engineering triangle, mathematical formula height.", "If you want to explore how this concept expands into other geometric contexts or practical applications, stay tuned—understanding altitude inverse proportionality unlocks deeper insight into shape efficiency and design stability."]
