A line that is also a perpendicular bisector of a line segment will pass through the midpoint of the line segment, and will have a slope that is equal to the negative reciprocal of the slope of the line segment (since they are perpendicular).
Hence, we simply have to solve for two things: i) the slope of the line segment, and ii) the midpoint of the line segment.
i) The slope of the segment is `m = (-8 - 4)/(3- (-3)) = (-12)/(6) = -2.`
ii) The midpoint of the segment has coordinates `x = (-3+3)/2 = 0` and `y = (4-8)/2 = -2` . Hence, the midpoint is (0, -2).
The slope of the line (perpendicular bisector) is the negative reciprocal of that of the line segment: `m_(l) = - (1/(-2)) = 1/2.`
Using the slope-intercept form:
`y - y_1 = m (x - x_1)`
`y - (-2) = 1/2 (x - 0)`
`y + 2 = 1/2 x`
`2y + 4 = x`
Hence, the equation of the line is `y = (x-4)/2` or `-x + 2y = -4` .