Question: Find the length of the longest altitude of a triangle with side lengths $ 7 $ cm, $ 10 $ cm, and $ 13 $ cm.
["Title: Find the Length of the Longest Altitude in a Triangle with Side Lengths 7 cm, 10 cm, and 13 cm", "When studying triangles in geometry, understanding altitudes is essential for calculating area, analyzing triangle properties, and solving real-world problems. One common challenge is determining the longest altitude—the height corresponding to the shortest side—which maximizes the vertical distance from a vertex to the opposite side. In this article, we’ll uncover how to find the length of the longest altitude for a triangle with side lengths 7 cm, 10 cm, and 13 cm—a scalene triangle with no equal sides—by combining triangle area computation and altitude formulas.", "---", "### What Is an Altitude in a Triangle?", "An altitude (or height) of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. For any triangle, the area ( A ) can be expressed in terms of any side ( b ) and its corresponding altitude ( h_b ):", "[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "Given three side lengths, ( a = 13\ \ ext{cm}, b = 10\ \ ext{cm}, c = 7\ \ ext{cm} ), we first determine the area using Heron’s formula, then use that area to compute each altitude. The longest altitude corresponds to the shortest side, since altitude decreases as base length increases for a fixed area.", "---", "### Step 1: Compute the Semi-Perimeter", "First, calculate the semi-perimeter ( s ) of the triangle:", "[\ns = \frac{a + b + c}{2} = \frac{13 + 10 + 7}{2} = \frac{30}{2} = 15\ \ ext{cm}\n]", "---", "### Step 2: Use Heron’s Formula to Find the Area", "Heron’s formula states the area ( A ) is:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "Substitute values:", "[\nA = \sqrt{15(15 - 13)(15 - 10)(15 - 7)} = \sqrt{15 \ imes 2 \ imes 5 \ imes 8}\n]", "[\nA = \sqrt{15 \ imes 2 \ imes 5 \ imes 8} = \sqrt{1200} = 20\sqrt{3}\ \ ext{cm}^2\n]", "(We keep it exact as ( 20\sqrt{3} ) for precision.)", "---", "### Step 3: Calculate Each Altitude", "Use the area formula rearranged to solve for each altitude:", "[\nh = \frac{2A}{\ ext{base}}\n]", "We compute the altitudes to each side:", "1. Altitude to side ( a = 13\ \ ext{cm} ):", "[\nh_{13} = \frac{2 \ imes 20\sqrt{3}}{13} = \frac{40\sqrt{3}}{13} \approx 5.33\ \ ext{cm}\n]", "2. Altitude to side ( b = 10\ \ ext{cm} ):", "[\nh_{10} = \frac{40\sqrt{3}}{10} = 4\sqrt{3} \approx 6.93\ \ ext{cm}\n]", "3. Altitude to side ( c = 7\ \ ext{cm} ) (shortest side, longest altitude expected):", "[\nh_7 = \frac{40\sqrt{3}}{7} \approx 9.79\ \ ext{cm}\n]", "---", "### Step 4: Identify the Longest Altitude", "Comparing the three:", "- ( h_{13} \approx 5.33\ \ ext{cm} )\n- ( h_{10} \approx 6.93\ \ ext{cm} )\n- ( h_7 \approx 9.79\ \ ext{cm} )", "Clearly, the longest altitude is to the shortest side (7 cm), and its length is:", "[\n\boxed{h_{\ ext{longest}} = \frac{40\sqrt{3}}{7} \ \ ext{cm}} \approx 9.79\ \ ext{cm}\n]", "---", "### Why This Matters", "Understanding the longest altitude helps in geometric visualization, engineering design, and optimization. Since altitude reflects perpendicular reach, maximizing it over different bases informs structural stability and spatial efficiency.", "---", "### Summary", "- Use Heron’s formula to compute area from side lengths.\n- Compute altitudes using ( h = \frac{2A}{\ ext{side}} ).\n- The longest altitude corresponds to the shortest side.\n- For triangle with sides 7 cm, 10 cm, 13 cm, the longest altitude is ( \frac{40\sqrt{3}}{7} \ \ ext{cm} ).", "This method applies to any scalene triangle—extract area via Heron’s formula, then calculate altitudes efficiently.", "---", "Keywords: longest altitude triangle, find triangle altitude, height of triangle with sides 7, 10, 13, Heron’s formula, altitude calculation, geometry problem solution", "Meta Description: Learn how to find the longest altitude of a triangle with side lengths 7 cm, 10 cm, and 13 cm using Heron’s formula and the area-to-height relationship. Step-by-step guide with exact values."]
