Thus, the area is reduced by \(\boxed{20\sqrt{3}}\) square centimeters.**Question:** A soil scientist is analyzing a plot of land shaped like a right triangle. The hypotenuse of the triangle is \(10\) meters, and the radius of the inscribed circle is \(2\) meters. Determine the ratio of the area of the inscribed circle to the area of the triangle.
["Title: Understanding the Area Ratio in a Right Triangle: A Soil Scientist’s Insight", "In this SEO-optimized article, we explore a real-world geometric scenario relevant to soil scientists analyzing shaped land plots. Consider a right triangle where the hypotenuse measures (10) meters and the inradius is (2) meters. We’ll determine the ratio between the area of the inscribed circle and the area of the triangle—key information for ecological planning and land management.", "---", "### Geometry Background", "For any right triangle, the inradius (r) is related to the legs (a), (b), and hypotenuse (c) by the formula:", "[\nr = \frac{a + b - c}{2}\n]", "Given:\n- Hypotenuse (c = 10) m\n- Inradius (r = 2) m", "Using the formula:", "[\n2 = \frac{a + b - 10}{2} \quad \Rightarrow \quad a + b - 10 = 4 \quad \Rightarrow \quad a + b = 14\n]", "---", "### Area of the Triangle", "The area (A) of a right triangle is:", "[\nA = \frac{1}{2}ab\n]", "We can also express the area in terms of the inradius and semiperimeter (s):", "[\nA = r \cdot s\n]", "First, compute the semiperimeter (s):", "[\ns = \frac{a + b + c}{2} = \frac{14 + 10}{2} = 12\n]", "Then:", "[\nA = r \cdot s = 2 \cdot 12 = 24 \ ext{ square meters}\n]", "---", "### Area of the Inscribed Circle", "The area of a circle with radius (r = 2) m is:", "[\nA_{\ ext{circle}} = \pi r^2 = \pi \cdot 2^2 = 4\pi \ ext{ square meters}\n]", "---", "### Ratio of Areas", "We now compute the ratio of the circle’s area to the triangle’s area:", "[\n\ ext{Ratio} = \frac{A_{\ ext{circle}}}{A_{\ ext{triangle}}} = \frac{4\pi}{24} = \frac{\pi}{6}\n]", "---", "### Final Answer and Summary", "Thus, the ratio of the area of the inscribed circle to the area of the triangle is:", "[\n\boxed{\frac{\pi}{6}}\n]", "This precise ratio helps soil scientists and environmental planners quantify space usage, drainage efficiency, and potential planting zones on triangular plots—highlighting how geometry supports sustainable land use.", "---", "Keywords: right triangle area ratio, inscribed circle area ratio, soil scientist land analysis, triangle geometry, ( \ ext{right triangle} ), ( r)-circle area, ( \frac{\pi}{6} ), land area calculation", "Meta Description: Calculate the ratio of the inscribed circle area to the triangle area in a right triangle with hypotenuse 10 m and inradius 2 m. Learn how geometry supports ecological planning with precise computation."]
