Indicate the equation of the line that is the perpendicular bisector of the segment with endpoints (4, 1) and (2, -5).

m = (-5 - 1)/(2-4) = 3

m' = -1/m = -1/3

y = -x/3 + b

find point on original line halfway between

x = (4 +2 )/2 = 3
y = (1 - 5)/2 = -2

so
-2 = -3/3 + b
b = -1
so
y = -x/3 - 1

3 y = -x - 3

To determine the equation of the perpendicular bisector of a line segment, we first need to find the midpoint of the segment and then determine the slope of the line through the two given points.

Step 1: Find the midpoint
The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) can be found using the midpoint formula:

x = (x₁ + x₂) / 2
y = (y₁ + y₂) / 2

In this case, the endpoints are (4, 1) and (2, -5). Let's substitute the values into the formula:

x = (4 + 2) / 2 = 6 / 2 = 3
y = (1 + -5) / 2 = -4 / 2 = -2

So, the midpoint is (3, -2).

Step 2: Determine the slope
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

slope = (y₂ - y₁) / (x₂ - x₁)

In this case, the two points are (4, 1) and (2, -5). Let's substitute the values into the formula:

slope = (-5 - 1) / (2 - 4) = -6 / -2 = 3

So, the slope of the line passing through the given points is 3.

Step 3: Determine the slope of the perpendicular bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of the line passing through the given points. In this case, the slope of the line is 3, so the slope of the perpendicular bisector would be -1/3.

Step 4: Write the equation of the line
The equation of a line with slope m and passing through a point (x₁, y₁) can be written in point-slope form as:

y - y₁ = m(x - x₁)

We already have the slope (-1/3) and the midpoint (3, -2). Substituting these values into the equation:

y - (-2) = (-1/3)(x - 3)
y + 2 = (-1/3)(x - 3)

Expanding and rearranging the equation:

y + 2 = (-1/3)x + 1
y = (-1/3)x - 1

So, the equation of the line that is the perpendicular bisector of the segment with endpoints (4, 1) and (2, -5) is y = (-1/3)x - 1.