In a right triangle with legs of lengths 9 cm and 12 cm, find the radius of the inscribed circle. Summarize the solution by providing the final result in a box.

["How to Find the Radius of the Inscribed Circle in a Right Triangle with Legs 9 cm and 12 cm", "In geometry, calculating the radius of the inscribed circle (also known as the incircle) in a right triangle is a common challenge. For a right triangle with legs of lengths 9 cm and 12 cm, there’s a straightforward method to determine this important measurement.", "First, recall that the triangle is right-angled, so the hypotenuse can be found using the Pythagorean theorem:", "[\nc = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \ ext{ cm}\n]", "The formula for the radius ( r ) of the inscribed circle in any triangle is:", "[\nr = \frac{A}{s}\n]", "where ( A ) is the area of the triangle and ( s ) is the semi-perimeter.", "Step 1: Compute the area\nThe area ( A ) of a right triangle is:", "[\nA = \frac{1}{2} \ imes \ ext{leg}_1 \ imes \ ext{leg}_2 = \frac{1}{2} \ imes 9 \ imes 12 = 54 \ ext{ cm}^2\n]", "Step 2: Compute the semi-perimeter\nThe perimeter is:", "[\n9 + 12 + 15 = 36 \ ext{ cm}\n]", "So the semi-perimeter ( s ) is:", "[\ns = \frac{36}{2} = 18 \ ext{ cm}\n]", "Step 3: Calculate the inradius\nNow plug into the formula:", "[\nr = \frac{A}{s} = \frac{54}{18} = 3 \ ext{ cm}\n]", "This elegant result shows that the radius of the inscribed circle in a right triangle with legs 9 cm and 12 cm is exactly 3 cm.", "Final Result:", "[\n\boxed{3 \ ext{ cm}}\n]"]