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📰 Question: A zoologist tracks an animal’s path forming a right triangle with hypotenuse $ 25 \, \text{m} $ and inradius $ 6 \, \text{m} $. What is the ratio of the inscribed circle’s area to the triangle’s area? 📰 Solution: For a right triangle, the inradius $ r = \frac{a + b - c}{2} $, where $ c $ is the hypotenuse. Let $ a $ and $ b $ be the legs. Then $ r = \frac{a + b - 25}{2} = 6 \Rightarrow a + b = 37 $. The area of the triangle is $ \frac{ab}{2} $, and the inradius formula $ r = \frac{a + b - c}{2} $ gives consistency. Using $ a^2 + b^2 = 625 $ and $ a + b = 37 $, solve for $ ab $: $ (a + b)^2 = 1369 = a^2 + b^2 + 2ab = 625 + 2ab \Rightarrow ab = 372 $. Area $ = 186 \, \text{m}^2 $. Circle area $ \pi r^2 = 36\pi $. Ratio $ \frac{36\pi}{186} = \boxed{\frac{6\pi}{31}} $. 📰 Question: A hydrologist calculates the angle of a groundwater flow slope using trigonometry. If the slope rises 1 meter over a horizontal distance of $ \sqrt{3} $ meters, what is $ \tan \theta $? 📰 The Ultimate Guide To Every Character In Spider Man Brand New Day Dont Miss These Hidden Gems 2250417 📰 Gh Metallica Cheats 6969413 📰 Down Home Fab Cast 4933582 📰 Var Resistor 5474362 📰 Add Cases With At Least Two Empty Teams 8898365 📰 What Angie Lorena Never Wants You To Know About Her Dark Past 3902722 📰 Batman Superman Enemies 263268 📰 What Is The Best Whole House Water Filter 7672187 📰 Discover The Romantic Charm Of Chinas Secret Cottagesno Tourist Crowds Allowed 7750228 📰 Wait Til You See These Sparkling Glass Christmas Ornamentsperfect For A Magical Holiday Experience 3072729 📰 What Does Unanimous Mean 5432554 📰 1989 Draft Nfl 8516046 📰 540 Pounding Pouty Pouty Secrets Why Every Woman Wishes She Had This Pout 1209683 📰 Shocking Facts About Brindle French Bulldogs You Must See Now 8578067 📰 Troubleshooting Verizon Wifi 3055694