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{4104} = \sqrt{4 \cdot 1026} = 2\sqrt{1026} $, but numerical approximation suffices here. Thus, the shortest altitude is $ \boxed{8.54} $ km.

Mon, 04 May 2026 23:43:49 +0000

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