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📅 May 2, 2026 👤 scraface

May 02, 2026

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

📌 A_1 = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \text{ square cm}

📌 If each side is increased by 2 cm, the new side length is \(8\) cm. The new area \(A_2\) is:

📌 A_2 = \frac{\sqrt{3}}{4} \times 8^2 = \frac{\sqrt{3}}{4} \times 64 = 16\sqrt{3} \text{ square cm}

📌 A_2 - A_1 = 16\sqrt{3} - 9\sqrt{3} = 7\sqrt{3} \text{ square cm}

📌 Thus, the area increases by \(\boxed{7\sqrt{3}}\) square cm.

📌 A rectangle with dimensions 3 cm by 4 cm is inscribed in a circle. What is the circumference of the circle?

📌 The diagonal of the rectangle is the diameter of the circle. Using the Pythagorean theorem, the diagonal \(d\) is:

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