help solving slope intercept form of line

Question

I need help on solving this question.

 

Find the equation in slope intercept form of the line that is the perpendicular bisector of the segment between (-3,4) and (3,-8) and list the formula.

 

thanks

Answer 1

OK, L, the first thing to do is to find the slope of the line on which the two points lie.

 

m = slope = [4 - (-8)]/{(-3) - 3] = 12/(-6) = -2

 

This means the line on which the points lie (let's call it L₁) has the general form

 

y = -2x + b₁ (we'll worry about b later).

 

Since the other line (we'll call it L₂) is perpendicular to L₁, it must, by definition, have a slope of (1/2), so it will have an equation of the general form

 

y = (x/2) + b₂

 

The midpoint of L₁ is given by (x2 + x1)/2 and (y2 + y1)/2 = [(-3) + 3)/2] and [4 + (-8)]/2 or (0, -2)

 

So L₂ will have a slope of (1/2) and pass through the point (0, -2)

 

This is all the information we need to calculate b₂.

 

y = (x/2) + b₂ ⇒ -2 = (0/2) + b₂

 

Therefore b₂ = -2, and the complete equation of L₂, the perpendicular bisector is:

 

y = (x/2) - 2

 

 

 

 

Answer 2

To work this problem we need to extract two things from the given line segment:  1) the midpoint and 2) the slope.  

 

To get the midpoint, we can use the midpoint formula:  midpoint = ( (x1+x2)/2 , (y1 + y2)/2 )

   This gives midpoint =   ( (-3 + 3)/2,  (4 + (-8))/2 ) =    (0, -2)

 

To get the slope we use   m =  rise/run =    (y2-y1)/(x2-x1)  

     This gives  m =  (-8 -4 )/ (3 - (-3))  =  -2.

 

The next step is to use the negative reciprocal rule to get the slope of the desired line.   The rule is that given a line with slope m,    any line perpendicular to it will have slope = -1/m   (the negative reciprocal)

    From this we see that the slope of the desired line is  - 1/(-2)  =  1/2

 

The final step is to use the point , slope formula to work out the equation of the desired line.  This formula is   y - y1 = m (x -x1).    For this formula (x1, y1) is the midpoint worked out above:  x1 = 0, y1  = -2

The slope for this formula is the slope worked out above:  m = 1/2 . Putting this together yields:

 

  y - (-2) = (1/2) (x - 0)

 

This can be rearrange to the standard slope intercept form:

 

  y =  (1/2) x - 2

 

Answer 3

We need to write an equation of a line in slope intercept form which is y= mx + b. So we need to know m (the slope) and b (the y intercept which is where the line crosses the y axis)

 

We know that the line we need the equation for is perpendicular to a given segment and also bisects the segment (meets it at the midpoint). The given segment runs between the two points (-3, 4) and (3, -8).

 

Step 1, find the midpoint of the given segment using the midpoint formula  M = ( x₁ + x₂  ,   y₁ + y_{2 )}

                                                                                                                          2               2

It doesn't matter which point we call point one so lets label our points  x₁  y₁     x₂  y₂

                                                                                                     (-3, 4)  (3 ,-8)

 

Plugging these values into the formula we get   M =  ( -3 + 3  , 4 + -8 )  which results in a midpoint of (0, -2)

                                                                                   2             2

 

So now we have a point for our line and in this example it happens to be the y intercept. You know this because the x value of the y intercept is always zero and the y value is b.

 

Step 2 we still need to find the slope of the given segment so we can use it to determine the slope of our line. Since the lines are perpendicular, we know the slope of our line will be the opposite inverse ( meaning flip the fraction over and change the sign) of the slope of the given segment.

 

slope   m = y₂ -y₁    using the points  x1  y1     x2  y2  we get  -8 - 4  =  -12   = -2

                  x₂ -x₁                                    (-3, 4)     (3 ,-8)               3- -3        6

 

Since our line is perpendicular, our slope will be the opposite inverse of -2   which is 1

                                                                                                         1               2

Now we just plug what we have found into the formula y =mx+b to get y= 1 X -2

                                                                                                            2