Find an equation for the perpendicular bisector of the line segment connecting the points (-2, 5) and (3, -7).

the slope of the line between these points is

(-7-5)/(3+2) = -12/5

so the slope of our bisector = -1/(-12/5)
= 5/12

so our perpendicular bisector is of form
y = (5/12) x + b

now it has to go through a point halfway between the given ones
average x = (-2+3)/2
average y = (5-7)/2
so (1/2 , -1 )
put that in our equation to find b
-1 = (5/12)(1/2) + b
-1 = 5/24 + b
b = -29/24
so
y = (5/12) x -29/24
y = (10/24)x - 29/24
24 y = 10 x - 29

To find an equation for the perpendicular bisector of a line segment, we'll need a few steps:

Step 1: Find the midpoint of the line segment.
Step 2: Determine the slope of the line containing the line segment.
Step 3: Find the negative reciprocal of the slope from step 2.
Step 4: Use the midpoint and the negative reciprocal slope to write the equation of the perpendicular bisector.

Let's go through these steps one by one to find the equation for the perpendicular bisector of the line segment connecting (-2, 5) and (3, -7).

Step 1: Finding the midpoint
To find the midpoint, we use the formula:
x = (x₁ + x₂)/2
y = (y₁ + y₂)/2

Given the points (-2, 5) and (3, -7), plugging the values into the formula, we get:
x = (-2 + 3)/2 = 1/2
y = (5 + -7)/2 = -2/2 = -1

Therefore, the midpoint is (1/2, -1).

Step 2: Finding the slope of the line segment
The slope of a line segment is given by the formula:
m = (y₂ - y₁)/(x₂ - x₁)

Plugging in the values from the two points, we have:
m = (-7 - 5)/(3 - (-2)) = -12/5

Step 3: Finding the negative reciprocal of the slope
The negative reciprocal of a slope is found by flipping the fraction and changing its sign. So, the negative reciprocal of -12/5 is 5/12.

Step 4: Writing the equation of the perpendicular bisector
Now that we have the midpoint (1/2, -1) and the negative reciprocal slope 5/12, we can use the point-slope form of an equation (y - y₁) = m(x - x₁) to write the equation.

Plugging in the values, we have:
(y - (-1)) = 5/12(x - 1/2)
(y + 1) = 5/12(x - 1/2)
y + 1 = (5x - 5/2)/12

By multiplying both sides by 12, we can get rid of the fraction:
12(y + 1) = 5x - 5/2
12y + 12 = 5x - 5/2
12y = 5x - 5/2 - 12
12y = 5x - 5/2 - 24/2
12y = 5x - 29/2

So, the equation of the perpendicular bisector is 12y = 5x - 29/2.