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Home » Evaluate the expression \( rac34 \cdot rac89 + rac13 - rac26 \). - NBX Soluciones

Evaluate the expression \( rac34 \cdot rac89 + rac13 - rac26 \). - NBX Soluciones

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 A_{\text{triangle}} = \frac{\sqrt{3}}{4} s^2

📌 For an equilateral triangle, the radius \( r \) of the inscribed circle is given by:

📌 r = \frac{s \sqrt{3}}{6}

📌 The area \( A_{\text{circle}} \) of the circle is:

📌 A_{\text{circle}} = \pi r^2 = \pi \left(\frac{s \sqrt{3}}{6}\right)^2 = \pi \frac{3s^2}{36} = \frac{\pi s^2}{12}

📌 Now, the ratio of the area of the circle to the area of the triangle is:

📌 \text{Ratio} = \frac{A_{\text{circle}}}{A_{\text{triangle}}} = \frac{\frac{\pi s^2}{12}}{\frac{\sqrt{3}}{4} s^2} = \frac{\pi s^2}{12} \cdot \frac{4}{\sqrt{3} s^2} = \frac{\pi}{3\sqrt{3}} = \frac{\pi \sqrt{3}}{9}

📌 Thus, the ratio of the area of the circle to the area of the triangle is \(\boxed{\frac{\pi \sqrt{3}}{9}}\).

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