# SOLUTION: What is the equation of the perpendicular bisector of the line between the points (2,2) and (6,6)?

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Question 183192: What is the equation of the perpendicular bisector of the line between the points (2,2) and (6,6)?
Found 2 solutions by Alan3354, jim_thompson5910:
You can put this solution on YOUR website!
What is the equation of the perpendicular bisector of the line between the points (2,2) and (6,6)?
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Find the point. The easiest way (always use the easiest) is to average x and y separately
(2+6)/2 = 4 (x and y in this case)
The mid-point is (4,4)
Now find the slope, m
m = diffy/diffx
m = (6-2)/(6-2)
m = 1
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The slope of lines perpendicular (there are an infinite number of them) have a slope that's the negative inverse.
m2 = -1
Then use
y-y1 = m2*(x-x1) where (x1,y1) is (4,4)
y-4 = -1(x-4)
y-4 = -x+4
x+y = 8

You can put this solution on YOUR website!
Step 1) First find midpoint of the points (2,2) and (6,6)

To find the midpoint, first we need to find the individual coordinates of the midpoint.

#### X-Coordinate of the Midpoint:

To find the x-coordinate of the midpoint, simply average the two x-coordinates of the given points by adding them up and dividing that result by 2 like this:

So the x-coordinate of the midpoint is

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#### Y-Coordinate of the Midpoint:

To find the y-coordinate of the midpoint, simply average the two y-coordinates of the given points by adding them up and dividing that result by 2 like this:

So the y-coordinate of the midpoint is

So the midpoint between the points and is

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Step 2) Find the slope of the line through the points (2,2) and (6,6)

Note: is the first point and is the second point .

Plug in , , , and

Subtract from to get

Subtract from to get

Reduce

So the slope of the line that goes through the points and is

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Step 3) Find the perpendicular slope

Take the slope and flip the fraction (think of it as ) to get and change the sign to get . So the perpendicular slope is

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Step 4) Find the equation of the line with the perpendicular slope (found in step 3) which goes through the midpoint (found in step 1)

To recap, the perpendicular slope is and the point that the perpendicular bisector goes through is (4,4)

So let's find the equation of the line with a slope and goes through the point (4,4)

If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation:

---Point-Slope Formula---
where is the slope, and is the given point

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)

Distribute

Multiply and to get

Add 4 to both sides to isolate y

Combine like terms and to get

So the equation of the line with a slope of which goes through the point (,) is:

which is now in form where the slope is and the y-intercept is

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