Question: A triangular geological fault line has side lengths of 10 cm, 13 cm, and 15 cm. Compute the length of the shortest altitude.
["Understanding Triangular Geological Fault Lines: Calculating the Shortest Altitude in a Triangle with Sides 10 cm, 13 cm, and 15 cm", "When analyzing geological features such as triangular fault lines, understanding key geometric properties is essential for risk assessment, structural modeling, and hazard analysis. One such crucial parameter is the shortest altitude of the triangular formation. In this article, we explore a specific geological triangular fault with side lengths of 10 cm, 13 cm, and 15 cm and provide a step-by-step solution to compute the shortest altitude.", "---", "### The Geometry of the Triangle", "Given a triangle with side lengths:\n- ( a = 10 , \ ext{cm} )\n- ( b = 13 , \ ext{cm} )\n- ( c = 15 , \ ext{cm} )", "First, verify that these side lengths form a valid triangle using the triangle inequality:", "- ( 10 + 13 > 15 ) → ( 23 > 15 ) ✅\n- ( 10 + 15 > 13 ) → ( 25 > 13 ) ✅\n- ( 13 + 15 > 10 ) → ( 28 > 10 ) ✅", "Since all conditions are satisfied, the triangle is valid.", "---", "### Why Compute the Shortest Altitude?", "In geological contexts, the altitude corresponding to the longest side (or "base") is often the shortest, especially in irregular fault boundaries. This influences how stress and strain distribute across faults, helping scientists model earthquake probabilities and structural weak points.", "---", "### Step-by-Step Calculation of the Shortest Altitude", "To find the shortest altitude, we compute the altitudes relative to each side in turn and identify the smallest.", "#### Step 1: Compute the Area Using Heron’s Formula", "First, calculate the semi-perimeter ( s ):", "[\ns = \frac{a + b + c}{2} = \frac{10 + 13 + 15}{2} = \frac{38}{2} = 19 , \ ext{cm}\n]", "Now apply Heron’s formula for area ( A ):", "[\nA = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{19(19 - 10)(19 - 13)(19 - 15)}\n]", "[\nA = \sqrt{19 \ imes 9 \ imes 6 \ imes 4} = \sqrt{19 \ imes 216}\n]", "[\nA = \sqrt{4104} \approx 64.07 , \ ext{cm}^2\n]", "---", "#### Step 2: Compute Altitudes Corresponding to Each Side", "The altitude ( h ) corresponding to a side ( x ) is given by:", "[\nh = \frac{2A}{x}\n]", "- Altitude to side ( a = 10 , \ ext{cm} ):", "[\nh_{10} = \frac{2 \ imes 64.07}{10} = \frac{128.14}{10} = 12.814 , \ ext{cm}\n]", "- Altitude to side ( b = 13 , \ ext{cm} ):", "[\nh_{13} = \frac{2 \ imes 64.07}{13} = \frac{128.14}{13} \approx 9.858 , \ ext{cm}\n]", "- Altitude to side ( c = 15 , \ ext{cm} ):", "[\nh_{15} = \frac{2 \ imes 64.07}{15} = \frac{128.14}{15} \approx 8.543 , \ ext{cm}\n]", "---", "### Conclusion: Identify the Shortest Altitude", "Comparing the three altitudes:", "- ( h_{10} \approx 12.814 , \ ext{cm} )\n- ( h_{13} \approx 9.858 , \ ext{cm} )\n- ( h_{15} \approx 8.543 , \ ext{cm} )", "The shortest altitude is approximately 8.543 cm, corresponding to the longest side (15 cm).", "---", "### Practical Implications in Geological Modeling", "In geological fault analysis, the shortest altitude highlights the minimal vertical divergence across the fault segment, which may indicate a high-stress concentration zone. Recognizing this enables researchers to prioritize monitoring and cataloging potential seismic activity along the most structurally vulnerable segment.", "---", "### Summary", "- Side lengths: 10 cm, 13 cm, 15 cm\n- Area: ( \approx 64.07 , \ ext{cm}^2 )\n- Shortest altitude: ( \approx 8.54 , \ ext{cm} ) (to the 15 cm side)", "Understanding these geometric parameters enhances the precision of geological hazard assessments and supports better predictions in fault dynamics.", "---", "Keywords: triangular geological fault, geological altitude calculation, fault line geometry, altitude of triangle, Heron’s formula, shortest altitude, rock mechanics, seismic hazard analysis", "---", "Next Steps: For field applications, using GPS and LiDAR data to precisely measure fault geometry improves the accuracy of such calculations, supporting earthquake preparedness and infrastructure planning."]
