Question: In a triangle with side lengths 13, 14, and 15 units, find the length of the longest altitude.
["# Exploring the Longest Altitude in a Triangle with Side Lengths 13, 14, and 15 Units", "In geometry, calculating the altitudes of a triangle provides valuable insight into its shape and area distribution. For triangles with sides measuring 13, 14, and 15 units—the famous 13-14-15 triangle—finding the longest altitude is both mathematically intriguing and practically useful. In this article, we’ll explore how to determine the longest altitude, why it occurs, and its length using precise geometric formulas.", "## Understanding the Triangle: 13, 14, 15", "The triangle with side lengths (a = 13), (b = 14), and (c = 15) is well-known in geometry due to its integer side lengths and ASA (Angle-Side-Angle) characteristics. These dimensions form a scalene triangle that satisfies the triangle inequality, ensuring a valid geometric shape.", "One remarkable property is that this triangle has an area calculable via Heron’s formula, which allows us to find height lengths relative to each side—key for identifying the longest altitude.", "## Step 1: Calculate the Area Using Heron’s Formula", "Heron’s formula states that the area (A) of a triangle with sides (a), (b), and (c) is:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "where (s) is the semi-perimeter:", "[\ns = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = 21\n]", "Now substitute into the formula:", "[\nA = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Simplify:", "[\nA = \sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} = 84\n]", "So, the area of the triangle is 84 square units.", "## Step 2: Find the Altitudes Using Area and Side Lengths", "The altitude (h_x) corresponding to side (x) is given by:", "[\nh_x = \frac{2A}{x}\n]", "Thus, compute the altitudes for each side:", "- Altitude to side 13:\n [\n h_{13} = \frac{2 \ imes 84}{13} = \frac{168}{13} \approx 12.92\n ]", "- Altitude to side 14:\n [\n h_{14} = \frac{168}{14} = 12\n ]", "- Altitude to side 15:\n [\n h_{15} = \frac{168}{15} = 11.2\n ]", "## Step 3: Identify the Longest Altitude", "Comparing:", "[\nh_{13} = \frac{168}{13} \approx 12.92,\quad h_{14} = 12,\quad h_{15} = 11.2\n]", "The longest altitude is (h_{13} = \frac{168}{13}), corresponding to the shortest side opposite the largest angle in the triangle.", "## Why Is the Longest Altitude on the Shortest Side?", "In any triangle, the longest altitude always drops to the shortest side. Since 13 is the shortest side among 13, 14, and 15, the altitude to the side of length 13 is the tallest. This principle aligns with the geometric intuition that an altitude increases as it falls perpendicularly to a shorter side within the same triangle.", "## Practical Applications", "Knowing the longest altitude helps in structural design, engineering stress analysis, and computer graphics where minimal distances between sides and vertices matter. For the 13-14-15 triangle, this value supports precise computations in related geometries.", "## Summary", "| Side | Altitude Length |\n|-------|---------------------|\n| 13 | ( \frac{168}{13} \approx 12.92 ) units (longest) |\n| 14 | 12 units |\n| 15 | 11.2 units |", "Final Answer: The longest altitude in the triangle with sides 13, 14, and 15 is ( \boxed{\frac{168}{13}} ) units, approximately 12.92 units. This occurs because the altitude to the shortest side is always the longest in a triangle.", "---", "## Want to Learn More?", "Explore further by visualizing the triangle, using coordinate geometry to compute altitudes, or studying how altitude lengths change with different triangles. Mastering these concepts deepens spatial reasoning and problem-solving skills in geometry."]
