Final answer:
The equation of the perpendicular bisector of the line segment with endpoints (1, -9) and (-9, -5) is y = (5/2)x + 3. This was found by calculating the midpoint, determining the slope of the original line, finding the negative reciprocal for the perpendicular slope, and then using the point-slope form of the equation.
Step-by-step explanation:
To find the equation of the perpendicular bisector of the line segment with endpoints (1, -9) and (-9, -5), we must first determine the midpoint of the segment as well as the slope of the perpendicular line.
The midpoint (M) can be found using the formula M = ((x1+x2)/2, (y1+y2)/2). Substituting the given points, we get M = ((1-9)/2, (-9-5)/2) which simplifies to M = (-4, -7).
The slope of the original line (m) is calculated using the end points: m = (y2-y1)/(x2-x1). This provides m = (-5 - (-9))/(-9 - 1) = 4/(-10) = -2/5. The slope of the perpendicular line will be the negative reciprocal of m, hence the slope of the perpendicular bisector (m') is 5/2.
Now, we use the point-slope form of the equation y - y1 = m'(x - x1) to find the equation of the perpendicular bisector. Substituting m' = 5/2 and the midpoint M = (-4, -7), we get the equation of the perpendicular bisector: y - (-7) = (5/2)(x - (-4)).
After simplifying, the equation becomes y + 7 = (5/2)x + 10, and then y = (5/2)x + 3. This is the equation of the perpendicular bisector.