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where is clemson university located - Coaching Toolbox

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 The area of the original equilateral triangle is \( A_1 = \frac{\sqrt{3}}{4} s^2 \). The area of the smaller equilateral triangle with side \( \frac{s}{2} \) is \( A_2 = \frac{\sqrt{3}}{4} \left(\frac{s}{2}\right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{s^2}{4} = \frac{\sqrt{3}}{16} s^2 \). The decrease in area is \( \Delta A = A_1 - A_2 = \frac{\sqrt{3}}{4} s^2 - \frac{\sqrt{3}}{16} s^2 = \left(\frac{4}{16} - \frac{1}{16}\right)\sqrt{3} s^2 = \frac{3\sqrt{3}}{16} s^2 \). The percentage decrease is:

📌 \frac{\Delta A}{A_1} \times 100 = \frac{\frac{3\sqrt{3}}{16} s^2}{\frac{\sqrt{3}}{4} s^2} \times 100 = \frac{3}{4} \times 100 = 75\%

📌 Thus, the percentage decrease in area is \(\boxed{75\%}\).

📌 This suite of questions and solutions integrates geometry with conceptual reasoning, appropriate for a math olympiad or advanced high school competition, while maintaining a clear and educational narrative.

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