Question: An aerospace engineer at Blue Origin is designing a triangular support beam with side lengths of 13 cm, 14 cm, and 15 cm. What is the length of the shortest altitude, in centimeters?

["An Aerospace Engineer at Blue Origin Is Designing a Triangular Support Beam — Here’s How to Find the Shortest Altitude", "When developing lightweight yet robust components for spaceflight, aerospace engineers often tackle intricate geometric challenges. One such challenge is determining critical structural properties—like altitude lengths—on optimized support components. In a recent innovatively designed triangular support beam by a Blue Origin engineer, the triangle features side lengths of 13 cm, 14 cm, and 15 cm. Understanding the shortest altitude of this triangle offers vital insight into its structural efficiency and load-bearing capability.", "### Understanding the Triangle: A 13-14-15 Triangle", "The 13-14-15 triangle is a well-known scalene triangle celebrated for its integer side lengths and notable geometric properties. Though not equilateral or isosceles, this triangle allows precise calculation of area and altitudes, crucial for engineering applications.", "#### Step 1: Compute the Area Using Heron’s Formula\nSince all sides are known, the first step is calculating the area using Heron’s formula.", "- Semi-perimeter:\n ( s = \frac{13 + 14 + 15}{2} = 21 ) cm", "- Area:\n [\n A = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)}\n ]\n [\n = \sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} = 84 \ ext{ cm}^2\n ]", "#### Step 2: Calculate Altitudes Corresponding to Each Side\nThe altitude of a triangle relative to a side is derived from the formula:\n[ h = \frac{2 \ imes \ ext{Area}}{\ ext{Base length}} ]", "- Altitude to side 13 cm:\n [\n h_{13} = \frac{2 \ imes 84}{13} = \frac{168}{13} \approx 12.92 \ ext{ cm}\n ]", "- Altitude to side 14 cm:\n [\n h_{14} = \frac{2 \ imes 84}{14} = \frac{168}{14} = 12 \ ext{ cm}\n ]", "- Altitude to side 15 cm:\n [\n h_{15} = \frac{2 \ imes 84}{15} = \frac{168}{15} = 11.2 \ ext{ cm}\n ]", "#### Step 3: Identify the Shortest Altitude\nComparing the three altitudes:\n- 12.92 cm (to 13 cm side),\n- 12 cm (to 14 cm side),\n- 11.2 cm (to 15 cm side)", "The shortest altitude is 11.2 cm.", "### Engineering Insight: Why This Matters in Aerospace Design", "In aerospace applications like rocket support structures, minimizing weight while maximizing strength and stability is essential. The shortest altitude corresponds to the longest side (the 15 cm side in this case), meaning this beam resists forces most effectively along its longest edge. Knowing that the altitude to the 15 cm side is only 11.2 cm indicates a compact yet strong design, crucial for high-efficiency rocket components.", "### Final Answer", "For the triangular support beam with sides 13 cm, 14 cm, and 15 cm, the shortest altitude is 11.2 cm—a critical dimension ensuring structural integrity in Blue Origin’s advanced aerospace engineering.", "---", "Optimizing triangular geometries through precise altitude calculations empowers engineers to create safer, lighter, and more efficient spacecraft. Understanding these principles continues to drive innovation at leading aerospace companies like Blue Origin."]