Final answer:
To find the equation of the median through point D in triangle DEF, calculate the midpoint of side EF, find the slope of the median, and use the slope-intercept form. The median is found to be a horizontal line with the equation y = -1.
Step-by-step explanation:
To find the equation of the median through point D(-5,-1) in triangle DEF, we first need to determine the midpoint of the opposite side. The opposite side is the line segment EF with endpoints E(3,3) and F(1,-5). The midpoint (M) can be found using the midpoint formula [(x1 + x2)/2, (y1 + y2)/2]. So for EF, the midpoint M is [(3+1)/2, (3+(-5))/2] = (2,-1).
Next, we calculate the slope of the median line, which is the change in y over the change in x between D and M. Using the slope formula (m = (y2 - y1)/(x2 - x1)), we find that the slope m is (-1 - (-1))/(2 - (-5)) = 0/7, which simplifies to m = 0. This suggests that the median is horizontal.
Since D has a y-coordinate of -1 and the median line is horizontal, the equation of the median is simply y = -1 in slope-intercept form, which is y = mx + b where m is the slope and b is the y-intercept.