To solve the problem, we need to find the equation of the median of a triangle. Here's the process:

1. Vertices of the triangle:

   • $A(4, 1)$
   • $B(-12, 8)$
   • $C(-2, -4)$

2. Midpoint of side BC: The median is drawn from vertex $A(4, 1)$ to the midpoint of side $BC$. So, we first need to find the midpoint of line segment $BC$.

   The formula for the midpoint of two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Applying this to points $B(-12, 8)$ and $C(-2, -4)$: $\text{Midpoint} = \left( \frac{-12 + (-2)}{2}, \frac{8 + (-4)}{2} \right) = \left( \frac{-14}{2}, \frac{4}{2} \right) = (-7, 2)$

3. Equation of the median: Now, the median is the line passing through points $A(4, 1)$ and the midpoint $(-7, 2)$. To find the equation of this line, we need to compute the slope between these two points.

   The slope $m$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Using points $A(4, 1)$ and $(-7, 2)$: $m = \frac{2 - 1}{-7 - 4} = \frac{1}{-11} = -\frac{1}{11}$

4. Equation of the line: Using the point-slope form of the line equation: $y - y_1 = m(x - x_1)$ Substituting $m = -\frac{1}{11}$ and point $A(4, 1)$: $y - 1 = -\frac{1}{11}(x - 4)$ Expanding this equation: $y - 1 = -\frac{1}{11}x + \frac{4}{11}$ $y = -\frac{1}{11}x + \frac{4}{11} + 1$ $y = -\frac{1}{11}x + \frac{4}{11} + \frac{11}{11} = -\frac{1}{11}x + \frac{15}{11}$

Thus, the equation of the median is: $y = -\frac{1}{11}x + \frac{15}{11}$ This equation is valid for the $x$-range between the x-coordinates of the points $A(4, 1)$ and the midpoint $(-7, 2)$, so $-7 \leq x \leq 4$.

Final Answer:

$y = -\frac{1}{11}x + \frac{15}{11} \quad \text{for} \quad -7 \leq x \leq 4$

Would you like further details or have any questions?

Here are some related questions:

1. How do you find the equation of a line given two points?
2. What is the geometric significance of the median in a triangle?
3. How do you calculate the midpoint of a line segment?
4. What is the point-slope form of the equation of a line?
5. Can a triangle have more than one median, and how are they related?

Tip: To check if two lines are parallel, compare their slopes. If they are equal, the lines are parallel!