Solution: First, compute the area of the triangle using Heron's formula. The semi-perimeter $ s = \frac{10 + 13 + 15}{2} = 19 $ km. The area $ A = \sqrt{19(19-10)(19-13)(19-15)} = \sqrt{19 \cdot 9 \cdot 6 \cdot 4} = \sqrt{4104} = 64.07 $ km² (approximate). The shortest altitude corresponds to the longest side (15 km). Using $ A = \frac{1}{2} \cdot \text{base} \cdot \text{height} $, the altitude $ h = \frac{2A}{15} = \frac{2 \cdot 64.07}{15} \approx 8.54 $ km. For exact value, simplify $ \sqrt{41

["How to Calculate the Shortest Altitude of a Triangle Using Heron’s Formula: A Step-by-Step Solution", "When tasked with finding the height of a triangle, especially the shortest one, using Heron’s formula is both efficient and precise. In this article, we walk through a clear solution to compute the shortest altitude—using a triangle with sides 10 km, 13 km, and 15 km—via a method reliable in geometry classrooms and engineering applications alike.", "---", "### Step 1: Compute the Semi-Perimeter", "Heron’s formula begins with the semi-perimeter ( s ), calculated as half the sum of the triangle’s sides:", "[\ns = \frac{a + b + c}{2} = \frac{10 + 13 + 15}{2} = 19 \ ext{ km}\n]", "This value plays a central role in determining the triangle’s area.", "---", "### Step 2: Compute the Area Using Heron’s Formula", "The area ( A ) of the triangle is given by:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "Substituting values:", "[\nA = \sqrt{19(19 - 10)(19 - 13)(19 - 15)} = \sqrt{19 \cdot 9 \cdot 6 \cdot 4}\n]", "Calculating the product inside the square root:", "[\n19 \cdot 9 = 171, \quad 6 \cdot 4 = 24, \quad 171 \cdot 24 = 4104\n]", "So,", "[\nA = \sqrt{4104} \approx 64.07 \ ext{ km}^2\n]", "Note: While we keep it exact here, a decimal approximation is often practical for practical engineering or arithmetic purposes.", "---", "### Step 3: Determine the Shortest Altitude", "In any triangle, the shortest altitude corresponds to the longest side—because altitude is inversely proportional to the base length for a fixed area. The longest side here is 15 km, so the shortest altitude will be perpendicular to this side.", "Using the area formula:", "[\nA = \frac{1}{2} \cdot \ ext{base} \cdot \ ext{height}\n]", "Solving for height ( h ):", "[\nh = \frac{2A}{\ ext{base}} = \frac{2 \cdot 64.07}{15} \approx 8.54 \ ext{ km}\n]", "---", "### Why This Method Works", "Heron’s formula ensures accurate area computation even for irregular triangles, and pairing it with the relationship between area, base, and height accurately identifies all altitudes. Rounding to two decimal places offers clarity without sacrificing precision in practical use.", "---", "### Final Result", "The shortest altitude of this triangle is approximately:", "[\n\boxed{8.54} \ ext{ km}\n]", "---", "This approach not only solves the problem but strengthens foundational skills in geometric computation—essential for students, architects, and data analysts alike. Use Heron’s formula confidently and compute altitudes with clarity."]