Find the equation in slope-intercept form of the line that is the perpendicular bisector of the segment between (-3,4) and (3,-8).

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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` .

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