Question: A historian discovers a triangular tablet with sides 17 cm, 25 cm, and 28 cm. What is the radius of the circumscribed circle around the triangle?
["Title: Discovering the Circumradius: How a Historian Calculated the Circle Ringing an Ancient Triangular Tablet", "In an astonishing archaeological find, a historian unearthed a triangular tablet measuring sides of 17 cm, 25 cm, and 28 cm—an almost perfect ancient artifact holding secrets of forgotten mathematics. The key question arising from this discovery isn’t just about the tablet’s physical dimensions, but a fascinating mathematical challenge: What is the radius of the circumscribed circle around the triangle?", "### Solving for the Circumscribed Circle (Circumradius)", "For any triangle with side lengths ( a ), ( b ), and ( c ), the radius ( R ) of the circumcircle—the circle that passes through all three vertices—is given by the formula:", "[\nR = \frac{abc}{4K}\n]", "where:\n- ( a ), ( b ), ( c ) are the side lengths,\n- ( K ) is the area of the triangle.", "#### Step 1: Check if it’s a valid triangle", "Before calculations, confirm the sides form a valid triangle using the triangle inequality:", "- ( 17 + 25 = 42 > 28 ) ✔️\n- ( 17 + 28 = 45 > 25 ) ✔️\n- ( 25 + 28 = 53 > 17 ) ✔️", "Valid triangle confirmed.", "#### Step 2: Compute the semi-perimeter ( s )", "[\ns = \frac{a + b + c}{2} = \frac{17 + 25 + 28}{2} = \frac{70}{2} = 35 \ ext{ cm}\n]", "#### Step 3: Calculate the area ( K ) using Heron’s formula", "[\nK = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{35(35 - 17)(35 - 25)(35 - 28)}\n]", "[\nK = \sqrt{35 \cdot 18 \cdot 10 \cdot 7}\n]", "Break down the product:", "- ( 35 = 5 \cdot 7 )\n- ( 18 = 2 \cdot 3^2 )\n- ( 10 = 2 \cdot 5 )\n- ( 7 = 7 )", "[\nK = \sqrt{(5 \cdot 7)(2 \cdot 3^2)(2 \cdot 5)(7)} = \sqrt{2^2 \cdot 3^2 \cdot 5^2 \cdot 7^2} = 2 \cdot 3 \cdot 5 \cdot 7 = 210 \ ext{ cm}^2\n]", "#### Step 4: Compute the circumradius ( R )", "Plug values into the circumradius formula:", "[\nR = \frac{abc}{4K} = \frac{17 \ imes 25 \ imes 28}{4 \ imes 210}\n]", "First calculate the numerator:", "[\n17 \ imes 25 = 425,\quad 425 \ imes 28 = 11,900\n]", "Then the denominator:", "[\n4 \ imes 210 = 840\n]", "Now divide:", "[\nR = \frac{11,900}{840} = \frac{11900 \div 20}{840 \div 20} = \frac{595}{42} = \frac{85}{6} \approx 14.17\ \ ext{cm}\n]", "Thus, the exact value is:", "[\nR = \frac{85}{6} \ ext{ cm}\n]", "### Why This Triangle Matters", "This triangular tablet not only speaks to ancient precision in geometry but also demonstrates how modern historians and mathematicians collaborate to reconstruct forgotten knowledge. The circumradius, a cornerstone of circle geometry, plays a vital role in fields ranging from architecture to astronomy. Finding ( R ) for this 17-25-28 triangle reveals both the beauty of mathematical truths and the enduring legacy of historical artifacts.", "### Final Answer:", "The radius of the circumscribed circle around the triangle is ( \frac{85}{6} ) cm, approximately 14.17 cm.", "---", "Keywords: circumradius formula, circumscribed circle circumference, triangle circumcircle, Heron’s formula, 17-25-28 triangle, archaeological math discovery, ancient geometry, circumradius calculation.\nMeta description: Discover how a triangular tablet with sides 17 cm, 25 cm, and 28 cm led a historian to calculate the radius of its circumscribed circle—expanding our understanding of ancient mathematical precision."]
