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?
["Understanding the Geometry: A Right Triangle with an Inscribed Circle", "In this fascinating problem, a cave formation is modeled geometrically as a right triangle with key measurements: a hypotenuse of 25 meters and an inscribed circle of radius 5 meters. We aim to find the ratio of the area of the inscribed circle to the area of the triangle, a classic problem blending geometry and algebra with real-world applications in natural cave formations.", "---", "### Step 1: Use Known Geometry Properties of Right Triangles", "Let the right triangle have legs $ a $ and $ b $, and hypotenuse $ c = 25 $. Since it’s a right triangle, we can use the inradius formula:", "For any right triangle, the radius $ r $ of the inscribed circle is:", "$$\nr = \frac{a + b - c}{2}\n$$", "Given $ r = 5 $ and $ c = 25 $, substitute:", "$$\n5 = \frac{a + b - 25}{2} \quad \Rightarrow \quad a + b - 25 = 10 \quad \Rightarrow \quad a + b = 35\n$$", "Also, by the Pythagorean Theorem:", "$$\na^2 + b^2 = c^2 = 25^2 = 625\n$$", "---", "### Step 2: Solve the System of Equations", "We now have:", "1. $ a + b = 35 $\n2. $ a^2 + b^2 = 625 $", "Use identity:", "$$\n(a + b)^2 = a^2 + b^2 + 2ab\n$$", "Substitute:", "$$\n35^2 = 625 + 2ab \quad \Rightarrow \quad 1225 = 625 + 2ab \quad \Rightarrow \quad 2ab = 600 \quad \Rightarrow \quad ab = 300\n$$", "---", "### Step 3: Compute Area of the Triangle", "Area $ A $ of the right triangle is:", "$$\nA = \frac{1}{2}ab = \frac{1}{2} \ imes 300 = 150 \ ext{ m}^2\n$$", "---", "### Step 4: Compute Area of the Inscribed Circle", "The area of a circle with radius $ r = 5 $ m is:", "$$\n\ ext{Area}_{\ ext{circle}} = \pi r^2 = \pi \ imes 25 = 25\pi \ ext{ m}^2\n$$", "---", "### Step 5: Compute the Ratio", "Ratio of the area of the circle to the area of the triangle:", "$$\n\ ext{Ratio} = \frac{25\pi}{150} = \frac{\pi}{6}\n$$", "---", "### Final Answer", "$$\n\boxed{\frac{\pi}{6}}\n$$", "This elegant ratio reveals not only a geometric truth but also enhances our understanding of natural cave formations modeled through precise mathematical principles. Whether in nature or design, such relationships help explain the beauty hidden within cave systems—formed through time, pressure, and the same geometric laws that shape our world."]
