1. The problem states a circle with equation $$(x - 2)^2 + (y - 4)^2 = 20$$ and asks if the point $(6, 6)$ lies on, inside, or outside the circle. 2. The general form of a circle's equation is $$(x - h)^2 + (y - k)^2 = r^2$$ where $(h, k)$ is the center and $r$ is the radius. 3. From the equation, the center is $(2, 4)$ and the radius is $$r = \sqrt{20} = 2\sqrt{5} \approx 4.472$$. 4. To determine the position of the point $(6, 6)$ relative to the circle, calculate the distance from the center to the point: $$d = \sqrt{(6 - 2)^2 + (6 - 4)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.472$$ 5. Compare the distance $d$ to the radius $r$: - If $d = r$, the point lies on the circle. - If $d < r$, the point lies inside the circle. - If $d > r$, the point lies outside the circle. 6. Since $d = r$, the point $(6, 6)$ lies exactly on the circle. **Final answer:** The point $(6, 6)$ lies on the circle.