Question: A triangle has side lengths 13 cm, 14 cm, and 15 cm. Find the length of the shortest altitude in centimeters.

["Finding the Shortest Altitude in a Triangle with Side Lengths 13 cm, 14 cm, and 15 cm", "When given a triangle with side lengths 13 cm, 14 cm, and 15 cm, one of the most insightful geometric questions involves calculating the shortest altitude. The altitude corresponding to each side varies depending on the triangle’s area, making it essential to understand how these altitudes relate to the triangle’s dimensions.", "In this article, we’ll explore how to determine the shortest altitude in a triangle with these precise side lengths.", "---", "### Why Calculate the Shortest Altitude?", "The altitude of a triangle is the perpendicular distance from a vertex to the opposite side. Because the area of the triangle remains constant, the shortest altitude corresponds to the longest side—since altitude is inversely proportional to the base length when area is fixed.", "This principle makes the shortest altitude especially interesting to compute and understand.", "---", "### Step 1: Use Heron’s Formula to Find the Area", "For a triangle with sides ( a = 13 ), ( b = 14 ), and ( c = 15 ), we start by calculating the semi-perimeter:", "[\ns = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \ ext{ cm}\n]", "Next, apply Heron’s formula:", "[\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)}\n]", "[\n= \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Calculating inside the square root:", "[\n21 \ imes 8 = 168, \quad 7 \ imes 6 = 42, \quad 168 \ imes 42 = 7056\n]", "[\n\ ext{Area} = \sqrt{7056} = 84 \ ext{ cm}^2\n]", "So, the area of the triangle is 84 cm².", "---", "### Step 2: Calculate Altitudes Using Area Formula", "The area ( A ) of a triangle is also given by:", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Rearranged to solve for height (altitude):", "[\n\ ext{height} = \frac{2 \ imes \ ext{Area}}{\ ext{base}}\n]", "We compute the altitudes corresponding to each side:", "- Altitude to side 13 cm:", "[\nh_{13} = \frac{2 \ imes 84}{13} = \frac{168}{13} \approx 12.92 \ ext{ cm}\n]", "- Altitude to side 14 cm:", "[\nh_{14} = \frac{168}{14} = 12 \ ext{ cm}\n]", "- Altitude to side 15 cm:", "[\nh_{15} = \frac{168}{15} = 11.2 \ ext{ cm}\n]", "---", "### Step 3: Identify the Shortest Altitude", "Comparing the three altitudes:", "- ( h_{13} \approx 12.92 ) cm\n- ( h_{14} = 12 ) cm\n- ( h_{15} = 11.2 ) cm", "The shortest altitude is therefore on the longest side, 15 cm, and measures:", "[\n\boxed{11.2 \ ext{ cm}}\n]", "---", "### Why This Method Matters", "This computation highlights an important geometry principle: smaller altitudes correspond to longer bases for fixed area. By leveraging Heron’s formula and the area formula, we efficiently determine precise altitude lengths and identify the shortest one.", "Whether in physics, engineering, or computer graphics, computing altitudes supports modeling triangular structures and analyzing forces, shadow lengths, and spatial relationships.", "---", "### Summary", "- Given triangle sides 13 cm, 14 cm, 15 cm\n- Semi-perimeter ( s = 21 ) cm\n- Area = 84 cm² (via Heron’s formula)\n- Altitudes:\n - To 13 cm side: 12.92 cm\n - To 14 cm side: 12 cm\n - To 15 cm side: 11.2 cm\n- Shortest altitude: 11.2 cm", "This elegant solution confirms that the shortest altitude in this triangle is exactly ( \boxed{11.2} ) cm."]