Final answer:
The equation of the median line from vertex A is y = -1/6x + 3.5. The equation of the perpendicular line passing through the foot of the altitude from vertex B can be found using the slope of the line passing through B and the midpoint of AC.
Step-by-step explanation:
To find the equation containing the median from vertex A, we need to first find the coordinates of the midpoint of the opposite side, which is BC. The midpoint, M, can be found by taking the average of the x-coordinates and the average of the y-coordinates of B and C. In this case, M is (-3, 5). Next, we need to find the equation of the line passing through A and M. We can use the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope. The slope can be found by taking the difference in y-coordinates divided by the difference in x-coordinates. Plugging in the values, we get y - 6 = (5 - 6)/(-3 - 3)(x - 3), which simplifies to y - 6 = -1/6(x - 3). Therefore, the equation of the median line from vertex A is y = -1/6x + 3.5. To find the equation containing the altitude from vertex B, we need to find the coordinates of the foot of the altitude, which is the point on the opposite side of the triangle that is perpendicular to B. In this case, the foot of the altitude, F, can be found by first finding the slope of the line passing through B and the midpoint of AC and then finding the equation of the perpendicular line passing through F. The slope of the line passing through B and the midpoint of AC can be found by taking the difference in y-coordinates divided by the difference in x-coordinates. Then, the slope of the perpendicular line is the negative reciprocal of the slope of the line passing through B and the midpoint of AC. Finally, using the point-slope form of a linear equation, we can find the equation of the perpendicular line passing through F.