Find the length of the shortest altitude of a triangle with side lengths \(13\) units, \(14\) units, and \(15\) units.
["Title: Find the Length of the Shortest Altitude of a Triangle with Side Lengths 13, 14, and 15 Units", "Meta Description:\nLearn how to calculate the shortest altitude of a triangle with sides 13, 14, and 15 units using Heron’s formula and area-based altitude formulas.", "---", "### Introduction", "Triangles with side lengths (a = 13), (b = 14), and (c = 15) units represent a well-known scalene triangle often studied in geometry. One of the most insightful properties of any triangle is the altitude — the perpendicular distance from a vertex to the opposite side — and identifying the shortest altitude can be critical for applications in physics, engineering, and design.", "In this article, we’ll explore how to calculate the length of the shortest altitude for a triangle with sides 13, 14, and 15 units using geometric principles and formulas.", "---", "### Understanding the Shortest Altitude", "In any triangle, the altitude corresponding to a side is inversely proportional to the length of that side — the longer the base, the shorter the altitude (assuming the area remains constant). Therefore, the shortest altitude belongs to the longest side.", "For sides 13, 14, and 15, the longest side is 15 units. Thus, the shortest altitude ((h_{15})) is the perpendicular distance from the opposite vertex to the side of length 15.", "---", "### Step 1: Compute the Area Using Heron’s Formula", "To find the altitude, we first compute the area of the triangle using Heron’s formula:", "[\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} = \frac{42}{2} = 21\n]", "Now apply the values into Heron’s formula:", "[\nA = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Calculate the product step-by-step:", "- (21 \ imes 8 = 168)\n- (168 \ imes 7 = 1176)\n- (1176 \ imes 6 = 7056)", "So:", "[\nA = \sqrt{7056} = 84 \ ext{ square units}\n]", "---", "### Step 2: Use Area to Find the Shortest Altitude", "The area of a triangle is also given by:", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Here, base = 15 units, and height = altitude to that base ((h_{15})).", "Rearranging:", "[\nh_{15} = \frac{2A}{\ ext{base}} = \frac{2 \ imes 84}{15} = \frac{168}{15} = 11.2 \ ext{ units}\n]", "---", "### Step 3: Verify Other Altitudes (Optional Confirmation)", "For completeness, compute all three altitudes to confirm 11.2 is the shortest.", "- Altitude to side 14: (h_{14} = \frac{2A}{14} = \frac{168}{14} = 12)\n- Altitude to side 13: (h_{13} = \frac{2A}{13} = \frac{168}{13} \approx 12.92)", "Indeed, (h_{15} = 11.2) is the shortest.", "---", "### Why This Matters", "Knowing the shortest altitude helps in:", "- Determining the triangle’s maximum height perpendicular to its longest side\n- Applications in construction, where minimal height affects stability and material use\n- Solving real-world geometry and trigonometry problems", "---", "### Conclusion", "The triangle with sides 13, 14, and 15 has its shortest altitude of 11.2 units, corresponding to the side of 15 units. This result follows naturally from Heron’s formula and the area-altitude relationship.", "For students, engineers, or enthusiasts, mastering such calculations builds a strong foundation in geometric reasoning and problem solving.", "---", "### SEO Keywords: \nShortest altitude of triangle 13-14-15, triangle altitude formula, Heron’s formula application, area of triangle 13-14-15, triangle geometry, physics triangles altitude, how to find altitude of a triangle, scalene triangle altitude.", "---", "### Bonus: Quick Formula Shortcut", "For any triangle, the shortest altitude can be found as:", "[\nh_{\ ext{shortest}} = \frac{2A}{\max(a,b,c)}\n]", "Since area (A = 84) and max side = 15, then:", "[\nh_{\ ext{shortest}} = \frac{2 \ imes 84}{15} = 11.2\n]", "Perfect for fast calculations!", "---", "Intrigued in deeper geometric proofs? Explore how altitudes relate to triangle area, trigonometry, or coordinates next."]
