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📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 Let the triangle have side lengths $ a = 7 $, $ b = 10 $, $ c = 13 $. First, compute the semi-perimeter $ s $:

📌 s = \frac{a + b + c}{2} = \frac{7 + 10 + 13}{2} = \frac{30}{2} = 15

📌 Now, use Heron’s formula to find the area $ A $:

📌 A = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{15(15 - 7)(15 - 10)(15 - 13)} = \sqrt{15 \cdot 8 \cdot 5 \cdot 2}

📌 A = \sqrt{15 \cdot 8 \cdot 10} = \sqrt{1200} = \sqrt{400 \cdot 3} = 20\sqrt{3}

📌 r = \frac{A}{s} = \frac{20\sqrt{3}}{15} = \frac{4\sqrt{3}}{3}

📌 Thus, the radius of the inscribed circle is $ \boxed{\dfrac{4\sqrt{3}}{3}} $ cm.

📌 Question: An entomologist observes a spiderweb forming a perfect equilateral triangle with side length 12 cm. What is the area of the circle that circumscribes this triangle?

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