For the triangle with sides 7, 8, 9, the shortest altitude (opposite side 9) is $\boxed{\dfrac{8\sqrt{5}}{3}}$.

["Finding the Shortest Altitude of a Triangle with Sides 7, 8, and 9", "In geometry, calculating the altitude corresponding to a given side helps us understand a triangle’s area and internal structure. For a triangle with side lengths 7, 8, and 9—where we seek the shortest altitude opposite the longest side (side 9)—we first compute the triangle’s area, then use the area to determine the altitude.", "### Step 1: Compute the Semi-Perimeter\nTo find the area, we use Heron’s Formula, starting by calculating the semi-perimeter $ s $:\n$$\ns = \dfrac{7 + 8 + 9}{2} = 12\n$$", "### Step 2: Apply Heron’s Formula for Area\nUsing Heron’s formula:\n$$\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{12(12 - 7)(12 - 8)(12 - 9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3}\n$$\n$$\n= \sqrt{720} = \sqrt{144 \cdot 5} = 12\sqrt{5}\n$$", "### Step 3: Compute the Altitude to Side 9\nThe altitude $ h $ opposite side 9 is found using the area formula:\n$$\n\ ext{Area} = \dfrac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n\Rightarrow 12\sqrt{5} = \dfrac{1}{2} \ imes 9 \ imes h\n$$\nSolving for $ h $:\n$$\nh = \dfrac{2 \ imes 12\sqrt{5}}{9} = \dfrac{24\sqrt{5}}{9} = \dfrac{8\sqrt{5}}{3}\n$$", "### Why is This the Shortest Altitude?\nThe shortest altitude in a triangle is always opposite the longest side. Since side 9 is the longest among 7, 8, and 9, the altitude to side 9 is the shortest.", "---", "Final Answer:\nThe shortest altitude, opposite side 9, is\n$$\n\boxed{\dfrac{8\sqrt{5}}{3}}\n$$"]