1. **Problem statement:** We have a triangle ABC with incentro I, area 12 cm², and perimeter 18 cm. A circle centered at I is tangent to side BC at point D. We need to: a) Name the circle. b) Calculate the radius of the circle in fraction form. c) Calculate the circumference length of the circle. 2. **Important formulas and concepts:** - The circle centered at the incentro I and tangent to the sides of the triangle is called the **incircle**. - The radius $r$ of the incircle is given by the formula: $$r = \frac{A}{s}$$ where $A$ is the area of the triangle and $s$ is the semiperimeter. - The semiperimeter $s$ is half the perimeter: $$s = \frac{P}{2}$$ - The circumference length $C$ of a circle is: $$C = 2 \pi r$$ 3. **Calculate the semiperimeter $s$:** $$s = \frac{18}{2} = 9$$ 4. **Calculate the radius $r$ of the incircle:** $$r = \frac{A}{s} = \frac{12}{9}$$ Simplify the fraction: $$r = \frac{\cancel{12}^{4}}{\cancel{9}^{3}} = \frac{4}{3}$$ 5. **Calculate the circumference length $C$:** $$C = 2 \pi r = 2 \pi \times \frac{4}{3} = \frac{8 \pi}{3}$$ **Final answers:** a) The circle is the **incircle** of the triangle. b) The radius of the incircle is $\frac{4}{3}$ cm. c) The circumference length is $\frac{8 \pi}{3}$ cm.