Final answer:
The slope of the line segment is -2, the midpoint is (2, -2), the slope of the perpendicular bisector is 1/2, and the equation of the perpendicular bisector in slope-intercept form is y = (1/2)x - 3.
Step-by-step explanation:
The student asked about the slope of a line segment with endpoints (4, -6) and (0, 2), the midpoint of the segment, the slope of the perpendicular bisector, and the equation of the perpendicular bisector in slope-intercept form.
To find the slope of the line segment, you subtract the y-coordinates and divide by the difference of the x-coordinates: (2 - (-6)) / (0 - 4) = 8 / -4 = -2. So, the slope is -2.
The midpoint of the line segment is calculated by taking the average of the x-coordinates and the y-coordinates: ((4 + 0) / 2, (-6 + 2) / 2) = (2, -2).
Since the slope of the original line is -2, the slope of the perpendicular bisector will be the negative reciprocal which is 1/2. To find the equation in slope-intercept form, use the midpoint as a point and the slope of 1/2: y - (-2) = (1/2)(x - 2), which simplifies to y = (1/2)x - 3.