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📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Original area:

📌 A_1 = \frac{\sqrt{3}}{4} \cdot 12^2 = \frac{\sqrt{3}}{4} \cdot 144 = 36\sqrt{3}

📌 New side length: $ 12 + 2 = 14 $ cm. New area:

📌 A_2 = \frac{\sqrt{3}}{4} \cdot 14^2 = \frac{\sqrt{3}}{4} \cdot 196 = 49\sqrt{3}

📌 \Delta A = A_2 - A_1 = 49\sqrt{3} - 36\sqrt{3} = 13\sqrt{3}

📌 Thus, the area increases by $ \boxed{13\sqrt{3}} $ square centimeters.

📌 Question: A cave formation is modeled as a right triangle with a hypotenuse of $ 25 $ m and an inscribed circle of radius $ 5 $ m. What is the ratio of the area of the circle to the area of the triangle?

📌 Solution: Let the hypotenuse be $ z = 25 $ m, and the inradius be $ c = 5 $ m. For a right triangle, the area $ A $ is given by:

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