1. **State the problem:** Find the length and midpoint of the line segment with endpoints $(-2, 6)$ and $(4, 0)$. 2. **Length formula:** The length $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate the differences:** $$x_2 - x_1 = 4 - (-2) = 4 + 2 = 6$$ $$y_2 - y_1 = 0 - 6 = -6$$ 4. **Substitute into the length formula:** $$d = \sqrt{6^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72}$$ 5. **Simplify the radical:** $$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$$ 6. **Midpoint formula:** The midpoint $M$ of the segment is: $$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$ 7. **Calculate the midpoint coordinates:** $$x_M = \frac{-2 + 4}{2} = \frac{2}{2} = 1$$ $$y_M = \frac{6 + 0}{2} = \frac{6}{2} = 3$$ **Final answers:** - Length of the line segment: $6\sqrt{2}$ - Midpoint of the line segment: $(1, 3)$