1. **Problem Statement:** Find the midpoint of a line segment given endpoints, and find the other endpoint given one endpoint and the midpoint. 2. **Formula for Midpoint:** The midpoint $M$ of a segment with endpoints $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by: $$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$ This means the midpoint's coordinates are the averages of the corresponding coordinates of the endpoints. 3. **Formula for Other Endpoint:** If the midpoint $M(x_m, y_m)$ and one endpoint $E(x_e, y_e)$ are known, the other endpoint $O(x_o, y_o)$ is: $$x_o = 2x_m - x_e$$ $$y_o = 2y_m - y_e$$ This comes from rearranging the midpoint formula. --- **A. Find Midpoints:** 1. For $P(-1, -6)$ and $R(-6, 5)$: $$x_m = \frac{-1 + (-6)}{2} = \frac{-7}{2} = -3.5$$ $$y_m = \frac{-6 + 5}{2} = \frac{-1}{2} = -0.5$$ Midpoint $M = (-3.5, -0.5)$ 2. For $W(-1.2, 1.0)$ and $A(5.2, -5.3)$: $$x_m = \frac{-1.2 + 5.2}{2} = \frac{4.0}{2} = 2.0$$ $$y_m = \frac{1.0 + (-5.3)}{2} = \frac{-4.3}{2} = -2.15$$ Midpoint $M = (2.0, -2.15)$ 3. For $C(2, -1)$ and $T(-6, 0)$: $$x_m = \frac{2 + (-6)}{2} = \frac{-4}{2} = -2$$ $$y_m = \frac{-1 + 0}{2} = \frac{-1}{2} = -0.5$$ Midpoint $M = (-2, -0.5)$ --- **B. Find Other Endpoints:** 1. Given endpoint $(2, 5)$ and midpoint $(5, 1)$: $$x_o = 2 \times 5 - 2 = 10 - 2 = 8$$ $$y_o = 2 \times 1 - 5 = 2 - 5 = -3$$ Other endpoint is $(8, -3)$ 2. Given endpoint $(-1, 9)$ and midpoint $(-9, -10)$: $$x_o = 2 \times (-9) - (-1) = -18 + 1 = -17$$ $$y_o = 2 \times (-10) - 9 = -20 - 9 = -29$$ Other endpoint is $(-17, -29)$