1. **Problem Statement:** We are given a triangular race track on a Cartesian plane with vertices at $(-2,0)$, $(2,0)$, and $(0,4)$. Each grid square represents an area of 9245 m². We need to find the exact length of the red track, which is the perimeter of the triangle. 2. **Formula and Rules:** The length of the track is the perimeter of the triangle, which is the sum of the lengths of its three sides. The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate the lengths of each side:** - Side between $(-2,0)$ and $(2,0)$: $$d_1 = \sqrt{(2 - (-2))^2 + (0 - 0)^2} = \sqrt{(4)^2 + 0} = \sqrt{16} = 4$$ - Side between $(2,0)$ and $(0,4)$: $$d_2 = \sqrt{(0 - 2)^2 + (4 - 0)^2} = \sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$$ - Side between $(0,4)$ and $(-2,0)$: $$d_3 = \sqrt{(-2 - 0)^2 + (0 - 4)^2} = \sqrt{(-2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$$ 4. **Sum the lengths to find the perimeter:** $$P = d_1 + d_2 + d_3 = 4 + 2\sqrt{5} + 2\sqrt{5} = 4 + 4\sqrt{5}$$ 5. **Interpretation:** The exact length of the red track in grid units is $4 + 4\sqrt{5}$. Since each grid square represents an area of 9245 m², the side lengths are in grid units, so the perimeter is also in grid units. To find the actual length in meters, we need the length of one grid unit side. However, the problem only asks for the exact length of the red track in terms of the grid units. **Final answer:** $$\boxed{4 + 4\sqrt{5}}$$