Solution: To find the shortest altitude, we first compute the area using Heron’s formula. Let the triangle have sides $a = 13$, $b = 14$, and $c = 15$. The semi-perimeter is:
["## Finding the Shortest Altitude: A Step-by-Step Solution Using Heron’s Formula", "When analyzing triangles, one important geometric quantity is the altitude—the perpendicular distance from a vertex to the opposite side. Among the three altitudes, the shortest altitude corresponds to the longest side, as altitude is inversely proportional to the base length for a fixed area. This article walks through a structured solution to find the shortest altitude of a triangle with sides ( a = 13 ), ( b = 14 ), and ( c = 15 ), using Heron’s formula to compute the area first.", "---", "### Step 1: Understand the Relationship Between Area and Altitudes", "The area of a triangle is given by:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "This means that the altitude ( h_a ) corresponding to side ( a ) is:", "[\nh_a = \frac{2 \ imes \ ext{Area}}{a}\n]", "Since altitude is inversely proportional to the base, the shortest altitude will be the one opposite the longest side.", "Given sides ( a = 13 ), ( b = 14 ), ( c = 15 ), the longest side is ( c = 15 ). Therefore, the shortest altitude is the one relative to side ( c ).", "---", "### Step 2: Compute the Semi-Perimeter", "To use Heron’s formula, we begin by computing the semi-perimeter ( s ):", "[\ns = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21\n]", "---", "### Step 3: Apply Heron’s Formula to Find the Area", "Heron’s formula states:", "[\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)}\n]", "Substituting the values:", "[\n\ ext{Area} = \sqrt{21 \ imes (21 - 13) \ imes (21 - 14) \ imes (21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Now compute the product inside the square root:", "[\n21 \ imes 8 = 168\n]\n[\n7 \ imes 6 = 42\n]\n[\n168 \ imes 42 = 7056\n]", "Thus:", "[\n\ ext{Area} = \sqrt{7056} = 84\n]", "---", "### Step 4: Calculate the Shortest Altitude", "Since the shortest altitude corresponds to the longest side ( c = 15 ), use:", "[\nh_c = \frac{2 \ imes \ ext{Area}}{c} = \frac{2 \ imes 84}{15} = \frac{168}{15} = 11.2\n]", "---", "### Conclusion", "Finding the shortest altitude in a triangle involves computing its area via Heron’s formula and identifying the longest side, as the altitude to that side will be the smallest. For triangle sides 13, 14, and 15:", "- Semi-perimeter: ( s = 21 )\n- Area: ( 84 )\n- Shortest altitude (to side ( c = 15 )): ( h_c = 11.2 )", "This method not only gives precision but also reveals how base length and area jointly determine altitudes—an essential insight in geometric analysis.", "---", "### Keywords:\n- Shortest altitude formula\n- Heron’s formula\n- Triangle area calculation\n- Altitude of a triangle\n- Longest side altitude\n- Geometry problem solving", "---", "Want to master triangle geometry? Learn more about Heron’s Formula, altitude calculation, and triangle properties at YourGeometryGuide.com."]
