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longfellow paul revere's ride - ECD Germany

📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 In a right triangle used to model seismic wave paths, the hypotenuse measures $d$ units, and the inradius of the triangle is $r$. If the wave reflects off the incircle center, forming a shortest path to a side, what is the ratio of the area of the incircle to the area of the triangle?

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

📌 The inradius $r$ of a right triangle is given by:

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

📌 But we want the ratio of the area of the incircle to the area of the triangle:

📌 But we can express this ratio more elegantly using known identities. Alternatively, recall:

📌 But instead of algebraic complexity, use a standard result: In a right triangle,

📌 However, the cleanest route is to use known geometric identities.

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