The altitude to a side is $ h = rac{2A}{ ext{base}} $. To find the longest altitude, use the shortest side, since altitude is inversely proportional to the base. The shortest side is 13, so:
["# Finding the Longest Altitude: Why the Shortest Side Matters", "In geometry, understanding the relationship between base length and altitude is crucial when solving problems involving triangle area and height. One key formula you’ll often encounter is:", "[ h = \frac{2A}{\ ext{base}} ]", "Where ( h ) is the altitude corresponding to a given base, and ( A ) is the area of the triangle. This formula shows a direct inverse relationship: the longer the base, the shorter the altitude, and vice versa.", "## The Strategy for Finding the Longest Altitude", "Since altitude ( h ) decreases as the base increases, to determine the longest altitude in a triangle, you should use the shortest side as the base. Why? Because dividing the fixed area ( A ) by a smaller base yields a larger altitude.", "This approach simplifies finding the maximum altitude—no need to calculate every possible altitude; just identify the shortest side and apply the formula.", "## Given Problem: The Shortest Side is 13", "The problem states that the shortest side of the triangle is ( 13 ) units. Using the altitude formula:", "[ h = \frac{2A}{13} ]", "Since ( A ) is fixed for the entire triangle, this expression shows that the altitude for the base of length 13 is the largest possible altitude in the triangle.", "## What This Means Geometrically", "The side of length ( 13 ) corresponds to the triangle’s smallest edge, and its opposite vertex has the highest point relative to that base—making its altitude the greatest. This is especially useful in optimization, engineering, and design, where maximizing height for a given base is often necessary.", "## Practical Example", "Suppose the area of a triangle is ( 39 ) square units, and its shortest side measures 13 units. Calculate the longest altitude:", "[ h = \frac{2 \ imes 39}{13} = \frac{78}{13} = 6 ]", "So, the altitude to the shortest side is 6 units—the longest altitude.", "## Conclusion", "To efficiently find the longest altitude in any triangle:\n1. Identify the shortest side.\n2. Apply ( h = \frac{2A}{\ ext{shortest side}} ).\n3. The result is the maximum height corresponding to the base.", "This method saves time, ensures correctness, and leverages the inverse proportionality central to triangle geometry.", "---", "Keywords: altitude formula, inversely proportional altitude, longest altitude triangle, shortest side altitude, triangle geometry, area and height relationship, 2A/base formula, geometry problem-solving", "Meta Description: Discover how to find the longest altitude in a triangle using the shortest side with the formula ( h = \frac{2A}{\ ext{base}} ). Learn why the smallest base gives the largest height in any triangle."]
