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Unlock Your PCs Full Power with the Fastest Microsoft Store Installation Method! - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 \boxed{8\pi x^2}

📌 Question: A quantum machine learning algorithm designer is analyzing a triangular lattice structure. If the triangle has sides of $ 5x $, $ 6x $, and $ 7x $ units, what is the radius of the inscribed circle in terms of $ x $?

📌 Let the sides of the triangle be $ a = 5x $, $ b = 6x $, and $ c = 7x $. First, compute the semi-perimeter:

📌 s = \frac{a + b + c}{2} = \frac{5x + 6x + 7x}{2} = 9x

📌 Use Heron’s formula to find the area $ A $:

📌 A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{9x(9x - 5x)(9x - 6x)(9x - 7x)} = \sqrt{9x(4x)(3x)(2x)} = \sqrt{216x^4} = 6x^2\sqrt{6}

📌 The radius $ r $ of the inscribed circle is given by:

📌 r = \frac{A}{s} = \frac{6x^2\sqrt{6}}{9x} = \frac{2x\sqrt{6}}{3}

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