Question 38557
<pre><font size = 5><b>The question is write an equation for the perpendicular
bisector of the line segment joining the two points?

the points are (0,0)(-8,-10)

1. First we find the midpoint between (0,0) and (-8,-10)
because the perpendicular bisector must pass through 
its midpoint.

2. Second we find the slope of the line through 
   (0,0) and (-8,-10)

3. Third we find the slope of the perpendicular bisector 
   by
   A. taking the reciprocal of the slope of the line 
      thru (0,0) and (-5,10)
   B. Multiplying this result by -1, which means 
      "changing the sign"

4. Fourth, we find the equation of the perpendicular 
   bisector using the point-slope form.

5. Simplify 

Here goes:

1. First we find the midpoint between (0,0) and (-8,-10)
because the perpendicular bisector must pass through 
its midpoint.

Midpoint = ( (x<sub>1</sub>+x<sub>2</sub>)/2, (y<sub>1</sub>+y<sub>2</sub>)/2 )

Midpoint = ( [0+(-8)]/2, [0+(-10)]/2 )

Midpoint = ( -8/2, -10/2 )

Midpoint = (-4,-5)  

2. Second we find the slope of the line through
   (0,0) and (-8,-10)

    y<sub>2</sub> - y<sub>1</sub>
m = ———————
    x<sub>2</sub> - x<sub>1</sub>

    (-10) - (0)
m = ————————————
     (-8) - (0)

m = (-10)/(-8)

m = 5/4

3. Third we find the slope of the perpendicular bisector 
   by
   A. taking the reciprocal of the slope of the line 
      thru (0,0) and (-5,10)
      
      reciprocal of 5/4 is 4/5

   B. Multiplying this result by -1, which means 
      "changing the sign"

      changing the sign of 4/5 we have -4/5

4. Fourth, we find the equation of the perpendicular 
   bisector using the point-slope form.

        y - y<sub>1</sub> = m(x - x<sub>1</sub>) where 

    m = -4/5 and (x<sub>1</sub>,y<sub>1</sub>) = Midpoint = (-4,-5)

      y - (-5) = -4/5[x - (-4) ]

5. Simplify

         y + 5 = -4/5(x + 4)

         y + 5 = -4/5x - 16/5

             y = -4/5x - 16/5 - 5

             y = -4/5x - 16/5 - 25/5

             y = -4/5x - 41/5

If you like you can put the equation in standard
form by clearing of fractions, getting x term first,
y term second, equal sign third and number fourth:

            5y = -4x - 41

       4x + 5y = -41

Edwin
AnlytcPhil@aol.com</pre>