Write the equation of the prpendicular bisector of the line segment AB with endpoints A(1,1) and B(7,5). Write the answer in slope-intercept form.

first you need the midpoint of the line segment

which is (4,3)
now find the slope of the given line which is 2/3
so the slope of the perpendicular is -3/2

equation:
y = (-3/2)x + b
but you know (4,3) lies on it , so
3= (-3/2)(4) + b
3 = -6 + b
b = 9

y = (-3/2)x + 9

To find the equation of the perpendicular bisector of a line segment, we need to follow these steps:

Step 1: Find the midpoint of the line segment AB.
Step 2: Find the slope of the line segment AB.
Step 3: Find the negative reciprocal of the slope from step 2 to get the slope of the perpendicular bisector.
Step 4: Use the slope from step 3 and the midpoint from step 1 to find the equation of the perpendicular bisector in slope-intercept form (y = mx + b).

Let's go through these steps:

Step 1: Find the midpoint of the line segment AB.
The coordinates of the midpoint (M) of line segment AB can be found by using the midpoint formula:

x-coordinate of M = (x-coordinate of A + x-coordinate of B) / 2
= (1 + 7) / 2
= 8 / 2
= 4

y-coordinate of M = (y-coordinate of A + y-coordinate of B) / 2
= (1 + 5) / 2
= 6 / 2
= 3

So, the midpoint of line segment AB is M(4,3).

Step 2: Find the slope of the line segment AB.
The slope (m) of a line can be found using the slope formula:

m = (y2 - y1) / (x2 - x1)

Let's label the coordinates of point A as (x1, y1) and the coordinates of point B as (x2, y2).
So, x1 = 1, y1 = 1, x2 = 7, and y2 = 5.

Now, we can calculate the slope:

m = (5 - 1) / (7 - 1)
= 4 / 6
= 2 / 3

So, the slope of line segment AB is 2/3.

Step 3: Find the negative reciprocal of the slope from step 2.
To find the slope of the perpendicular bisector, we need to take the negative reciprocal of the slope of AB.

Negative reciprocal of 2/3 = -3/2

So, the slope of the perpendicular bisector is -3/2.

Step 4: Use the slope from step 3 and the midpoint from step 1 to find the equation of the perpendicular bisector in slope-intercept form (y = mx + b).

Using the slope-intercept form:

y = mx + b

Substituting the slope (-3/2) and the coordinates of the midpoint (4,3):

3 = (-3/2) * 4 + b

Simplifying:

3 = -6 + b

Add 6 to both sides:

9 = b

So, the y-intercept (b) of the perpendicular bisector is 9.

Therefore, the equation of the perpendicular bisector of line segment AB in slope-intercept form is:

y = (-3/2)x + 9