The altitude corresponding to side $a$ is $h_a = \frac{2A}{a} = \frac{168}{13} \approx 12.92$, for $b$ it is $h_b = \frac{168}{14} = 12$, and for $c$ it is $h_c = \frac{168}{15} = 11.2$. The shortest altitude corresponds to the longest side, which is $15$, yielding:

["Understanding Altitudes in a Triangle: A Step-by-Step Breakdown with Numerical Examples", "In geometry, understanding the relationships between a triangle’s sides and its altitudes is essential for solving problems in trigonometry, area calculations, and optimization. One key insight is that the shortest altitude corresponds to the longest side—a principle rooted in the inverse relationship between side length and altitude height when area is held constant.", "### What Are Altitudes in a Triangle?", "An altitude of a triangle is a perpendicular segment from a vertex to the opposite side (or its extension). For a triangle with area ( A ) and side lengths ( a, b, c ), the altitudes are determined by:", "[\nh_a = \frac{2A}{a}, \quad h_b = \frac{2A}{b}, \quad h_c = \frac{2A}{c}\n]", "Since the area ( A ) is fixed for a given triangle, larger side lengths result in smaller altitudes—a direct consequence of this formula.", "---", "### Given Numbers and Derived Altitudes", "In this example, the total area ( A ) is given as ( 168 ) square units. Using this, we calculate the altitudes for sides ( a = 13 ), ( b = 14 ), and ( c = 15 ):", "- Altitude on side ( a = 13 ):\n [\n h_a = \frac{2 \ imes 168}{13} = \frac{336}{13} \approx 25.85\n ]\n Wait — this contradicts the provided value ( h_a = \frac{168}{13} \approx 12.92 ). Let’s clarify.", "Wait — the formula in the prompt simplifies directly to ( h = \frac{2A}{\ ext{side}} ), which suggests a unit consistency or perhaps the area was normalized. Let’s reconcile the values.", "Actually, there’s a standard identity: the area ( A = \sqrt{s(s-a)(s-b)(s-c)} ), but here the formula presents altitudes in terms simply as ( h = \frac{2A}{\ ext{side}} ), which matches ( A = \frac{1}{2} a h_a \Rightarrow h_a = \frac{2A}{a} ), so ( 2A ) in the numerator implies the full expression accounts for the constant. Thus, the values:", "- ( h_a = \frac{168}{13} \approx 12.92 ) — this suggests ( 2A = 168 ), so ( A = 84 ).\n- Similarly, ( h_b = \frac{168}{14} = 12 ), ( h_c = \frac{168}{15} = 11.2 ).", "Thus, interpreting ( 2A = 168 \Rightarrow A = 84 ) is consistent.", "Therefore, the altitudes are:", "- ( h_a = \frac{168}{13} \approx 12.92 )\n- ( h_b = \frac{168}{14} = 12 )\n- ( h_c = \frac{168}{15} = 11.2 )", "---", "### The Shortest Altitude: Shortest Side, Tallest Height", "Since ( c = 15 ) is the longest side, it produces the smallest altitude:\n[\nh_c = \frac{168}{15} = 11.2\n]", "This confirms the geometric principle: the shortest altitude always corresponds to the longest side, maximizing perpendicularity and minimizing length.", "Comparing:", "- ( h_a \approx 12.92 )\n- ( h_b = 12 )\n- ( h_c = 11.2 )", "So,\n[\n\boxed{h_c = \frac{168}{15} = 11.2}\n]\nis the shortest altitude, consistent with topological expectations.", "---", "### Why This Matters", "This relationship is crucial in:", "- Designing stable structures where minimal height under load matters\n- Computational geometry tasks like bounding box optimization\n- Problem-solving strategies in competition math and engineering applications", "By identifying which side is longest, you instantly identify the shortest altitude—saving time and computation.", "---", "Key Takeaway:\nThe altitude length inversely scales with side length. The longest side encloses the shortest perpendicular distance (altitude), and packets of geometry knowledge like this reveal elegant, predictable patterns across all similar triangles. For any triangle with area ( A = 168 ), the altitudes compute directly—so verify numerical inputs, but when ( c = 15 ), ( h_c = 11.2 ) is indeed the shortest.", "---", "Understanding how area, side length, and altitude interrelate empowers faster, more accurate geometric reasoning—start analyzing triangles with confidence."]