5**Question:** A palynologist is studying a triangular pollen grain with sides measuring 13 cm, 14 cm, and 15 cm. Determine the length of the shortest altitude of this triangle.

["Title: Shortest Altitude of a Triangular Pollen Grain: A Step-by-Step Analysis Using a 13–14–15 Triangle", "Scientists studying microscopic structures often turn to geometry for insight—and few shapes are as studied in mathematics as the triangle. In this article, we delve into a fascinating case: a triangular pollen grain with side lengths 13 cm, 14 cm, and 15 cm. Using precise geometric methods, we determine the length of the shortest altitude in this triangle—critical for understanding its structural and functional properties in nature.", "---", "### The Triangle Behind the Study: 13–14–15 Triangle", "The pollen grain’s shape, examined under high magnification, forms a scalene triangle with sides of 13 cm, 14 cm, and 15 cm. This particular triangle is well known in geometry for being heronian—meaning it has integer side lengths and integer area—making it ideal for detailed measurement.", "---", "### Step 1: Calculate the Area Using Heron’s Formula", "To find the altitude, we first need the area. For any triangle with sides (a), (b), and (c), Heron’s formula computes the area as:", "[\ns = \frac{a + b + c}{2} \quad \ ext{(semi-perimeter)}\n]\n[\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)}\n]", "Apply values: (a = 13), (b = 14), (c = 15)\nSemi-perimeter:\n[\ns = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21~\ ext{cm}\n]", "Now calculate:\n[\n\ ext{Area} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "[\n= \sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} = 84~\ ext{cm}^2\n]", "The area of the pollen grain triangle is 84 square centimeters.", "---", "### Step 2: Use Area to Find Each Altitude", "Altitude ((h)) of a triangle is related to area ((A)) and base ((b)) by:\n[\nA = \frac{1}{2} \ imes b \ imes h \quad \Rightarrow \quad h = \frac{2A}{b}\n]", "The three altitudes correspond to the three sides:\n- Altitude to side 13 cm: (h_{13} = \frac{2 \ imes 84}{13} = \frac{168}{13} \approx 12.92~\ ext{cm})\n- Altitude to side 14 cm: (h_{14} = \frac{2 \ imes 84}{14} = \frac{168}{14} = 12~\ ext{cm})\n- Altitude to side 15 cm: (h_{15} = \frac{2 \ imes 84}{15} = \frac{168}{15} = 11.2~\ ext{cm})", "---", "### Step 3: Identify the Shortest Altitude", "Comparing the altitudes:\n- (h_{13} \approx 12.92~\ ext{cm})\n- (h_{14} = 12~\ ext{cm})\n- (h_{15} = 11.2~\ ext{cm})", "The shortest altitude is to the longest side—15 cm—measuring 11.2 cm.", "---", "### Why This Matters in Palynology", "Understanding the geometry and internal dimensions of pollen grains aids researchers in identifying species, reconstructing ancient ecosystems, and exploring mechanical resilience. The shortest altitude to the 15 cm side reveals the point of least structural support, offering clues about how the grain interacts with air, water, or surfaces during dispersal.", "---", "### Summary", "For a triangular pollen grain with sides 13 cm, 14 cm, and 15 cm:\n- Area = 84 cm²\n- Shortest altitude = 11.2 cm (to the 15 cm side)", "This geometric insight exemplifies how mathematical precision deepens our understanding of nature’s microscopic design.", "---", "Keywords: palynology, triangular pollen grain, shortest altitude, Heron’s formula, 13–14–15 triangle, science education, geometry in nature, pollen structure analysis."]