Final answer:
The equation of the line through the perpendicular bisectors of the given line segment is y = -2x + 15, which is not one of the listed options.
Step-by-step explanation:
To write the equation of the line through the perpendicular bisectors of the line segment connecting the points (4, 2) and (8, 4), we must first find the midpoint and the slope of the line segment.
The midpoint, M, is the average of the x-coordinates and the y-coordinates of the two points: M = ((4 + 8)/2, (2 + 4)/2) = (6, 3).
The slope of the line segment, m, between the two points is (4 - 2) / (8 - 4) = 2/4 = 1/2. The slope of the perpendicular bisector will be the negative reciprocal of m, so the slope of the perpendicular bisector is -2.
Now we have a point (6, 3) and a slope -2 for the perpendicular bisector. Using the slope-intercept form y = mx + b, we can substitute our known values to find b: 3 = -2(6) + b, which simplifies to b = 3 + 12 = 15.
Therefore, the equation of the perpendicular bisector is y = -2x + 15, which is not listed among the choices given a), b), c), or d).