1. The problem is to analyze the set of points given: $\{(-6,0), (9,6), (0,-2), (-3,-5), (-6,-7), (5,7)\}$.\n\n2. Since these are discrete points, we can discuss their coordinates, distances, or any patterns. Here, we will calculate the distance between the first two points as an example.\n\n3. The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$\n\n4. Using points $(-6,0)$ and $(9,6)$: $$d = \sqrt{(9 - (-6))^2 + (6 - 0)^2} = \sqrt{(9 + 6)^2 + 6^2} = \sqrt{15^2 + 6^2} = \sqrt{225 + 36} = \sqrt{261}.$$\n\n5. Simplify $\sqrt{261}$ if possible. Since $261 = 9 \times 29$, we have: $$\sqrt{261} = \sqrt{9 \times 29} = \sqrt{9} \times \sqrt{29} = 3\sqrt{29}.$$\n\n6. Therefore, the distance between points $(-6,0)$ and $(9,6)$ is $3\sqrt{29}$.\n\nThis method can be applied to find distances between any pairs of points in the set.