Let the legs of the right triangle be \(a\) and \(b\), and the hypotenuse \(c = 10\). The radius \(r\) of the inscribed circle is given by:

["Understanding the Inscribed Circle Radius in a Right Triangle with Hypotenuse (c = 10)", "In geometry, right triangles offer fascinating relationships between side lengths and special properties like the radius of the inscribed circle. One key formula connects the inradius (r) to the legs (a) and (b) of a right triangle and its hypotenuse (c):", "[\nr = \frac{a + b - c}{2}\n]", "This article explores how this formula works when the hypotenuse is fixed at (c = 10), and explains how to determine (a) and (b) while maximizing or analyzing the inradius.", "---", "### The Formula for the Inradius of a Right Triangle", "For any right triangle with perpendicular legs (a) and (b), and hypotenuse (c), the radius of the incircle (the circle tangent to all three sides) is:", "[\nr = \frac{a + b - c}{2}\n]", "Since (c = 10), the formula simplifies to:", "[\nr = \frac{a + b - 10}{2}\n]", "This elegant relation arises from combining the area-based definition of the inradius with the Pythagorean Theorem.", "---", "### Using the Pythagorean Theorem", "By the Pythagorean Theorem:", "[\na^2 + b^2 = c^2 = 100\n]", "Together with the inradius formula, we have a system:", "[\n\begin{cases}\na + b = 2r + 10 \\na^2 + b^2 = 100\n\end{cases}\n]", "Instead of solving directly for (r), this system helps find how connected (a), (b), and (r) are. In fact, eliminating (a) and (b) reveals that (a + b = 2r + 10), so increasing (r) requires a larger sum (a + b), constrained only by the fixed (c = 10).", "---", "### Maximizing the Inradius (r)", "To find the maximum possible radius (r), note that (a + b) is maximized when the triangle is isosceles, i.e., (a = b), due to the symmetry and efficiency of the isosceles right triangle.", "Set (a = b). Then:", "[\na^2 + a^2 = 100 \Rightarrow 2a^2 = 100 \Rightarrow a^2 = 50 \Rightarrow a = \sqrt{50} = 5\sqrt{2}\n]", "So, (a = b = 5\sqrt{2}), and:", "[\na + b = 10\sqrt{2}\n]", "Now compute (r):", "[\nr = \frac{a + b - c}{2} = \frac{10\sqrt{2} - 10}{2} = 5(\sqrt{2} - 1)\n]", "Approximately:", "[\nr \approx 5(1.4142 - 1) = 5(0.4142) = 2.071\n]", "Thus, the maximum inradius occurs when the triangle is isosceles with (r = 5(\sqrt{2} - 1)).", "---", "### How (r) Changes with Different (a) and (b)", "For any (a, b > 0) such that (a^2 + b^2 = 100), the radius (r = \frac{a + b - 10}{2}) increases as (a + b) increases.", "Using algebra, the sum (a + b) can be maximized under the constraint (a^2 + b^2 = 100). By the Cauchy-Schwarz inequality or Lagrange multipliers, the maximum sum (a + b) under (a^2 + b^2 = 100) is indeed (10\sqrt{2}), achieved uniquely (up to order) when (a = b = 5\sqrt{2}).", "Therefore:", "[\nr_{\ ext{max}} = \max r = \frac{10\sqrt{2} - 10}{2} = 5(\sqrt{2} - 1)\n]", "---", "### Applications and Why It Matters", "Understanding (r) in a right triangle with fixed hypotenuse helps in:", "- Designing structures with maximal inscribed circle (e.g., efficient circulation in rounded corners),\n- Solving optimization problems in geometry and engineering,\n- Teaching fundamental relationships between triangle elements.", "The formula (r = \frac{a + b - c}{2}) is not just a mathematical curiosity—it’s a gateway to deeper geometric insight.", "---", "### Summary", "- The inradius (r) of a right triangle with hypotenuse (c = 10) is (r = \frac{a + b - 10}{2}).\n- The sum (a + b) is maximized when (a = b = 5\sqrt{2}), yielding maximum (r = 5(\sqrt{2} - 1)).\n- This relationship connects area, perimeter, and inradius in elegant ways, underscoring the harmony in Euclidean geometry.", "For anyone studying triangles, the inscribed circle remains a cornerstone concept—especially when guided by simple yet powerful formulas like (r = \frac{a + b - c}{2}).", "---", "Keywords:\nright triangle, inradius formula, inscribed circle right triangle, inscribed circle radius, (r = \frac{a + b - c}{2}), hypotenuse (c = 10), isosceles right triangle, geometry problem, triangle optimization, triangle properties", "Meta Description:\nDiscover how the inradius (r) of a right triangle with hypotenuse 10 relates to its legs (a) and (b), how to maximize (r), and why the formula (r = \frac{a + b - c}{2}) is fundamental in geometry."]