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Home » Divide numerator and denominator by \(x^2\): - inBeat

Divide numerator and denominator by \(x^2\): - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Question:** A retired scientist is mentoring students on geometric proportions. An equilateral triangle has an area of \( 36\sqrt{3} \, \text{cm}^2 \). If each side is reduced by 2 cm, by how many square centimeters does the area decrease?

📌 Set the area equal to \( 36\sqrt{3} \):

📌 \frac{1}{4} s^2 = 36 \Rightarrow s^2 = 144 \Rightarrow s = 12 \, \text{cm}

📌 New side length after reduction:

📌 s' = 12 - 2 = 10 \, \text{cm}

📌 A' = \frac{\sqrt{3}}{4} (10)^2 = \frac{\sqrt{3}}{4} \times 100 = 25\sqrt{3} \, \text{cm}^2

📌 \Delta A = 36\sqrt{3} - 25\sqrt{3} = 11\sqrt{3} \, \text{cm}^2

📌 \boxed{11\sqrt{3} \, \text{cm}^2}

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