Solution: The area of the triangle is $ \frac{1}{2} \times 9 \times 12 = 54 $ km². The altitudes corresponding to each side are $ \frac{2 \times 54}{9} = 12 $, $ \frac{2 \times 54}{12} = 9 $, and $ \frac{2 \times 54}{15} = 7.2 $. The shortest altitude is $ 7.2 $ km, or $ \frac{36}{5} $. Thus, the length is $\boxed{\dfrac{36}{5}}$ km.

["Solution: Finding the Shortest Altitude of a Triangle Using Area and Side Lengths", "Understanding the relationship between a triangle’s area, side lengths, and altitudes is essential in geometry. This SEO-optimized guide explains how to calculate the shortest altitude using a well-known triangle area formula, making it easier for students, educators, and math enthusiasts to efficiently solve altitude problems.", "---", "Understanding Triangle Area and Altitudes", "The area ( A ) of a triangle can be calculated using the formula:\n[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]\nRearranging this, the altitude corresponding to a given side is:\n[\nh = \frac{2A}{\ ext{side length}}\n]", "In this example, the triangle has an area of ( 54 ) km² — computed from one side measuring 9 km and its corresponding altitude of 12 km, or another 12 km and a 9 km base. This confirms the consistency of the area value.", "---", "Step-by-step Calculation of Altitudes", "1. Altitude corresponding to the 9 km side:\n[\nh_1 = \frac{2 \ imes 54}{9} = \frac{108}{9} = 12 \ ext{ km}\n]", "2. Altitude corresponding to the 12 km side:\n[\nh_2 = \frac{2 \ imes 54}{12} = \frac{108}{12} = 9 \ ext{ km}\n]", "3. Altitude corresponding to the 15 km side (derived from hypotenuse using Pythagoras):\nGiven the triangle is right-angled (9–12–15 Pythagorean triple), the hypotenuse is 15 km:\n[\nh_3 = \frac{2 \ imes 54}{15} = \frac{108}{15} = 7.2 \ ext{ km}\n]", "---", "Identifying the Shortest Altitude", "Comparing the three altitudes:\n- 12 km\n- 9 km\n- 7.2 km", "The shortest altitude is clearly ( 7.2 ) km, which can also be expressed exactly as:\n[\n\boxed{\dfrac{36}{5}} \ ext{ km}\n]", "---", "Why This Solution Matters", "This elegant method showcases how area-based reasoning simplifies altitude calculations. Whether in textbook problems, exam prep, or real-life applications like land surveying, using area as the anchor point ensures accuracy and efficiency. The shortest altitude often reveals insightful geometric properties and supports optimization strategies.", "---", "Key Takeaway\nWith the base and area known, the altitude is found directly — and when side lengths form a Pythagorean triple like 9–12–15, calculations become intuitive. Use ( h = \frac{2A}{a} ) for every side to count altitudes quickly and confidently.", "---", "Final Answer:\nThe shortest altitude is (\boxed{\dfrac{36}{5}}) km."]