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Home » Thus, the total number of distinct labeling arrangements is \(\boxed243\). - Sourci

Thus, the total number of distinct labeling arrangements is \(\boxed243\). - Sourci

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 The original area \(A_1\) of an equilateral triangle with side \(s = 10\) is:

📌 A_1 = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 10^2 = 25\sqrt{3}

📌 The new side length is \(12\) cm, so the new area \(A_2\) is:

📌 A_2 = \frac{\sqrt{3}}{4} \times 12^2 = 36\sqrt{3}

📌 \Delta A = A_2 - A_1 = 36\sqrt{3} - 25\sqrt{3} = 11\sqrt{3}

📌 Thus, the area increases by \(\boxed{11\sqrt{3}}\) square centimeters.

📌 Compute \(\cos 60^\circ\) using the unit circle and verify using the properties of an equilateral triangle.

📌 On the unit circle, the coordinates of the point corresponding to \(60^\circ\) are \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). Thus, \(\cos 60^\circ = \frac{1}{2}\).

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