1. **State the problem:** Find the midpoint of the missing term and calculate the equation of each point of vertices given points T(-4,6), P(2,4), A(5,10), E(0,0), and points S, R, Q with vector notations near the triangle. 2. **Find the midpoint of segment T and A:** The midpoint formula is $$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ For T(-4,6) and A(5,10): $$x_m = \frac{-4+5}{2} = \frac{1}{2} = 0.5$$ $$y_m = \frac{6+10}{2} = \frac{16}{2} = 8$$ Midpoint M = (0.5, 8) 3. **Calculate the equation of the line through points T(-4,6) and A(5,10):** Slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 6}{5 - (-4)} = \frac{4}{9}$$ Using point-slope form: $$y - y_1 = m(x - x_1)$$ $$y - 6 = \frac{4}{9}(x + 4)$$ Simplify: $$y - 6 = \frac{4}{9}x + \frac{16}{9}$$ $$y = \frac{4}{9}x + \frac{16}{9} + 6 = \frac{4}{9}x + \frac{16}{9} + \frac{54}{9} = \frac{4}{9}x + \frac{70}{9}$$ 4. **Calculate the equation of the line through points E(0,0) and P(2,4):** Slope: $$m = \frac{4 - 0}{2 - 0} = 2$$ Equation: $$y - 0 = 2(x - 0)$$ $$y = 2x$$ 5. **Calculate the equation of the line through points A(5,10) and E(0,0):** Slope: $$m = \frac{10 - 0}{5 - 0} = 2$$ Equation: $$y - 0 = 2(x - 0)$$ $$y = 2x$$ 6. **Summary:** - Midpoint of T and A is (0.5, 8). - Equation of line TA: $$y = \frac{4}{9}x + \frac{70}{9}$$ - Equation of line EP: $$y = 2x$$ - Equation of line AE: $$y = 2x$$ Since points S, R, Q and vectors A̅F̅, K̅T̅, S̅A̅ are not fully defined with coordinates, their equations cannot be calculated here. This completes the solution for the first problem.