1. **State the problem:** We are given the vertices of triangle PQR as $P(3,4)$, $Q(7,-2)$, and $R(-2,-1)$. We need to find the equation of the median through vertex $R$. 2. **Recall the definition of a median:** A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. 3. **Find the midpoint of side $PQ$:** The midpoint $M$ of segment $PQ$ is given by the midpoint formula: $$M = \left(\frac{x_P + x_Q}{2}, \frac{y_P + y_Q}{2}\right)$$ Substitute the coordinates: $$M = \left(\frac{3 + 7}{2}, \frac{4 + (-2)}{2}\right) = \left(\frac{10}{2}, \frac{2}{2}\right) = (5,1)$$ 4. **Find the slope of the median $RM$:** The slope $m$ between points $R(-2,-1)$ and $M(5,1)$ is: $$m = \frac{y_M - y_R}{x_M - x_R} = \frac{1 - (-1)}{5 - (-2)} = \frac{2}{7}$$ 5. **Write the equation of the median line through $R$ with slope $m$:** Using point-slope form: $$y - y_1 = m(x - x_1)$$ Substitute $m=\frac{2}{7}$ and point $R(-2,-1)$: $$y - (-1) = \frac{2}{7}(x - (-2))$$ $$y + 1 = \frac{2}{7}(x + 2)$$ 6. **Simplify the equation:** $$y + 1 = \frac{2}{7}x + \frac{4}{7}$$ $$y = \frac{2}{7}x + \frac{4}{7} - 1 = \frac{2}{7}x - \frac{3}{7}$$ **Final answer:** The equation of the median through vertex $R$ is $$y = \frac{2}{7}x - \frac{3}{7}$$