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

it shows a series of boxes and says to fill them in... _+_ _=_ _ _ _ _
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slope = [ -5 - 1 ] / [ 2 - 4 ]

= -6/-2 = 3 = m in y = m x + b
so
y = 3 x + b
put in (4,1)
1 = 3 (4) + b
1 = 12 + b
b = -11
so
y = 3 x - 11
NOW find line through the middle and perpendicular
middle x = (1/2)(4+2)= 3
middle y = (1/2)(1-5) = -2
so through (3,-2)
slope = -1/3
so
y = -(1/3)x + b
-2 = -(1/3)(3) + b
-2 = -1 + b
b = -1
so
y = -(1/3)x - 1
3 y = -x -3
x+3y = -3

3.65^(1/3)

(3650/1000)^(1/3)
(1/10)(3650)^(1/3)
hhmmm try 12 nope
try 15 -->225*15 -->3375 close
so about 1.5
try1.55
3.72, not going to get much closer than 1.55 without calculator

To find the equation of the line that is the perpendicular bisector of the segment with endpoints (4, 1) and (2, -5), we can follow these steps:

Step 1: Find the mid-point of the segment:
The mid-point formula is given by:
Mid-point (x, y) = ((x1 + x2) / 2, (y1 + y2) / 2)

Using the coordinates (4, 1) and (2, -5):
Mid-point (x, y) = ((4 + 2) / 2, (1 + -5) / 2)
Mid-point (x, y) = (6 / 2, -4 / 2)
Mid-point (x, y) = (3, -2)

So, the mid-point is (3, -2).

Step 2: Find the slope of the given line segment:
The slope of a line segment between two points (x1, y1) and (x2, y2) is given by:
Slope (m) = (y2 - y1) / (x2 - x1)

Using the coordinates (4, 1) and (2, -5):
Slope (m) = (-5 - 1) / (2 - 4)
Slope (m) = (-6) / (-2)
Slope (m) = 3

So, the slope of the given line segment between the endpoints (4, 1) and (2, -5) is 3.

Step 3: Find the negative reciprocal of the slope:
Since we want to find the slope of the perpendicular bisector, we need to find the negative reciprocal of the slope found in Step 2.

Negative reciprocal = -1 / m

Negative reciprocal = -1 / 3

So, the negative reciprocal of the slope is -1/3.

Step 4: Write the equation of the line using the mid-point and the negative reciprocal slope.

The equation of the line is in the form y = mx + b, where m is the slope and b is the y-intercept. Since we know the mid-point and the slope, we can use the mid-point coordinates and substitute them into the equation.

Using the mid-point (x, y) = (3, -2) and the negative reciprocal slope m = -1/3, the equation becomes:

y = -1/3x + b

Substituting the mid-point into the equation:
-2 = (-1/3)(3) + b
-2 = -1 + b
b = -2 + 1
b = -1

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

Not GOOD AT ALL I JUST NEEDED THEANSWER