Find the length of the shortest altitude in a triangle with sides measuring 13, 14, and 15 units.
["# Find the Length of the Shortest Altitude in a Triangle with Sides 13, 14, and 15 Units", "Triangles with side lengths 13, 14, and 15 units form a well-known triangle often used in geometry problems due to their integer side lengths and interesting properties. This scalene triangle is not only mathematically significant but also a favorite in altitude-related exercises. In this article, we will explore how to determine the shortest altitude of a triangle with sides 13, 14, and 15 and calculate its exact length.", "---", "## Understanding Altitudes in a Triangle", "An altitude of a triangle is a perpendicular segment from a vertex to the opposite side (or its extension). The altitude lengths depend on both side lengths and the area of the triangle. For a triangle with known sides and area, the altitude from a vertex to a side is calculated using the area formula:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} \quad \Rightarrow \quad \ ext{height} = \frac{2 \ imes \ ext{Area}}{\ ext{base}}\n]", "Hence, the shortest altitude corresponds to the longest side, as the height is inversely proportional to the base.", "---", "## Step 1: Calculate the Area Using Heron’s Formula", "Given sides (a = 13), (b = 14), and (c = 15):", "1. Compute the semi-perimeter:\n[\ns = \frac{13 + 14 + 15}{2} = 21\n]", "2. Apply Heron’s formula for the area (A):\n[\nA = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Calculate the product inside the square root:\n[\n21 \ imes 8 = 168,\quad 7 \ imes 6 = 42,\quad 168 \ imes 42 = 7056\n]", "[\nA = \sqrt{7056} = 84\n]", "So, the area of the triangle is 84 square units.", "---", "## Step 2: Find Altitudes Corresponding to Each Side", "Now calculate the altitude from each vertex to the opposite side using:\n[\nh = \frac{2A}{\ ext{side}}\n]", "- Altitude to side 13:\n[\nh_{13} = \frac{2 \ imes 84}{13} = \frac{168}{13} \approx 12.923\n]", "- Altitude to side 14:\n[\nh_{14} = \frac{168}{14} = 12\n]", "- Altitude to side 15:\n[\nh_{15} = \frac{168}{15} = 11.2\n]", "---", "## Step 3: Identify the Shortest Altitude", "From the calculations:\n- (h_{13} \approx 12.923)\n- (h_{14} = 12)\n- (h_{15} = 11.2)", "The shortest altitude is (h_{15} = 11.2) units, corresponding to the longest side (15 units).", "---", "## Why the Shortest Altitude Is Important", "The shortest altitude reflects how close the triangle’s area is concentrated along the longest side. This property has applications in engineering, architecture, and physics where load distribution along triangular supports matters.", "---", "## Summary", "| Side | Altitude Length (units) |\n|--------|-------------------------|\n| 13 | ( \frac{168}{13} \approx 12.923 ) (longest base) |\n| 14 | 12 |\n| 15 | ( 11.2 = \frac{56}{5} ) (shortest altitude: shortest) |", "Conclusion: The shortest altitude in a triangle with sides 13, 14, and 15 units measures 11.2 units, calculated as (\frac{168}{15} = \frac{56}{5}).", "---", "## Final Notes", "- For scalene triangles, the shortest altitude is always opposite the longest side.\n- Always use Heron’s formula to compute area when sides are given but no right angle is assumed.\n- Understanding altitudes helps solve real-world geometry problems efficiently.", "---", "Keywords: shortest altitude triangle sides 13 14 15, find altitude in triangle, area and altitude formula, Heron’s formula 13–14–15 triangle, triangle geometry, altitudes calculation."]
