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a serious house on serious earth - AssociationVoting

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 Solution: Let the side length of the original equilateral triangle be $ s $. The area $ A $ of an equilateral triangle is given by:

📌 Question: In a smart grid layout modeled as a right triangle, the hypotenuse measures $ z $ meters and the inradius (radius of the inscribed circle) is $ c $ meters. Express the ratio of the area of the inscribed circle to the area of the triangle in terms of $ c $ and $ z $.

📌 Solution: Let the right triangle have legs $ a $, $ b $, and hypotenuse $ z $. The area $ A $ of the triangle is:

📌 Also, the area can be expressed in terms of the inradius and semiperimeter $ s $:

📌 But for a right triangle, there is a known identity:

📌 We want the ratio of the area of the inscribed circle to the area of the triangle:

📌 Now express $ s $ in terms of $ c $ and $ z $. From the inradius formula:

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