#### 30000Question: In a right triangle, the radius of the inscribed circle is $ r $, and the hypotenuse is $ h $. If the two legs are in the ratio $ 3:4 $, find the ratio of the area of the circle to the area of the triangle in terms of $ r $.
["Title: Area of Inscribed Circle vs. Triangle Area: A 3:4:5 Right Triangle", "In right triangles, understanding the relationship between the inradius, leg proportions, and overall area reveals elegant geometric insights. This article explores a special case: a right triangle with legs in the ratio 3:4, hypotenuse $ h $, and inradius $ r $. We’ll determine the ratio of the area of the inscribed circle to the area of the triangle, expressed solely in terms of $ r $.", "---", "### Setting the Triangle", "Let the legs of the right triangle be $ 3x $ and $ 4x $. Then, by the Pythagorean Theorem:", "$$\nh = \sqrt{(3x)^2 + (4x)^2} = \sqrt{9x^2 + 16x^2} = \sqrt{25x^2} = 5x\n$$", "So the sides are:\n- Leg A = $ 3x $\n- Leg B = $ 4x $\n- Hypotenuse $ h = 5x $", "---", "### Inradius of a Right Triangle", "For any right triangle with legs $ a $, $ b $, and hypotenuse $ c $, the inradius $ r $ is given by:", "$$\nr = \frac{a + b - c}{2}\n$$", "Substituting $ a = 3x $, $ b = 4x $, $ c = 5x $:", "$$\nr = \frac{3x + 4x - 5x}{2} = \frac{2x}{2} = x\n$$", "Thus, $ x = r $", "---", "### Area of the Triangle", "The area $ A_{\ riangle} $ is:", "$$\nA_{\ riangle} = \frac{1}{2} \ imes 3x \ imes 4x = \frac{1}{2} \cdot 12x^2 = 6x^2\n$$", "Substituting $ x = r $:", "$$\nA_{\ riangle} = 6r^2\n$$", "---", "### Area of the Inscribed Circle", "The circle inscribed in a triangle has area:", "$$\nA_{\ ext{circle}} = \pi r^2\n$$", "---", "### Ratio of Areas", "Now compute the ratio:", "$$\n\frac{A_{\ ext{circle}}}{A_{\ riangle}} = \frac{\pi r^2}{6r^2} = \frac{\pi}{6}\n$$", "---", "### Final Insight", "Despite the variable leg ratio and hypotenuse, when the legs are in a 3:4 proportion forming a Pythagorean triple, the inradius simplifies neatly to $ r = x $, and the area ratio becomes a constant—independent of the actual size—because both areas scale with $ r^2 $.", "Thus, the ratio of the area of the inscribed circle to the area of the triangle is:", "$$\n\boxed{\frac{\pi}{6}}\n$$", "a beautiful and universal result in geometric design.", "---", "Keywords: inradius of right triangle, inscribed circle area ratio, 3-4-5 triangle geometry, triangle area in terms of inradius, ratio of circle to triangle area."]
