Question 183192
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.

<h4>X-Coordinate of the Midpoint:</h4>

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:

{{{x[mid]=(2+6)/2=8/2=4}}}

So the x-coordinate of the midpoint is {{{x=4}}}

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<h4>Y-Coordinate of the Midpoint:</h4>

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:

{{{y[mid]=(2+6)/2=8/2=4}}}

So the y-coordinate of the midpoint is {{{y=4}}}

So the midpoint between the points *[Tex \LARGE \left(2,2\right)] and *[Tex \LARGE \left(6,6\right)] is *[Tex \LARGE \left(4,4\right)]

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

Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(2,2\right)] and *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(6,6\right)].

{{{m=(6-2)/(6-2)}}} Plug in {{{y[2]=6}}}, {{{y[1]=2}}}, {{{x[2]=6}}}, and {{{x[1]=2}}}

{{{m=(4)/(6-2)}}} Subtract {{{2}}} from {{{6}}} to get {{{4}}}

{{{m=(4)/(4)}}} Subtract {{{2}}} from {{{6}}} to get {{{4}}}

{{{m=1}}} Reduce

So the slope of the line that goes through the points *[Tex \LARGE \left(2,2\right)] and *[Tex \LARGE \left(6,6\right)] is {{{m=1}}}

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Step 3) Find the perpendicular slope

Take the slope {{{m=1}}} and flip the fraction (think of it as {{{m=1/1}}}) to get {{{m=1/1}}} and change the sign to get {{{m=-1/1}}}. So the perpendicular slope is {{{m=-1}}}

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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 {{{m=-1}}} and the point that the perpendicular bisector goes through is (4,4)

So let's find the equation of the line with a slope {{{m=-1}}} and goes through the point (4,4)

If you want to find the equation of line with a given a slope of {{{-1}}} which goes through the point ({{{4}}},{{{4}}}), you can simply use the point-slope formula to find the equation:

---Point-Slope Formula---
{{{y-y[1]=m(x-x[1])}}} where {{{m}}} is the slope, and *[Tex \Large \left(x_{1},y_{1}\right)] is the given point

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

{{{y-4=(-1)(x-4)}}} Plug in {{{m=-1}}}, {{{x[1]=4}}}, and {{{y[1]=4}}} (these values are given)

{{{y-4=-x+(-1)(-4)}}} Distribute {{{-1}}}

{{{y-4=-x+4}}} Multiply {{{-1}}} and {{{-4}}} to get {{{4}}}

{{{y=-x+4+4}}} Add 4 to  both sides to isolate y

{{{y=-x+8}}} Combine like terms {{{4}}} and {{{4}}} to get {{{8}}}

So the equation of the line with a slope of {{{-1}}} which goes through the point ({{{4}}},{{{4}}}) is:

{{{y=-x+8}}} which is now in {{{y=mx+b}}} form where the slope is {{{m=-1}}} and the y-intercept is {{{b=8}}}

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So the equation of the perpendicular bisector of the line between the points (2,2) and (6,6) is {{{y=-x+8}}}

So the answer you're looking for is {{{y=-x+8}}}

Here's the graph to verify the answer:

<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/perpendicularbisector.png">