Question

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

Answer 1

Answer:

The correct answer is x+3y=-3

Step-by-step explanation:

My fist step to solving this question would be to find the mid-point of the given line. 

The mid point of (4,1) and (2,-5) is (3,-2). The mid point is where the perpendicular bisector connects or bisects the given segment. 

My second step would be to graph the two given points and to connect them, forming a line. This way, I would know the slope of the line and then I would be able to find the slope of the perpendicular bisector, since the slope for perpendicular lines is the opposite reciprocal of the given line.

In doing this, I discovered that the slope of the segment with the given endpoints is 3 which means that the slope of the perpendicular bisector will be x

So, so far we've got a point of intersection and a slope which is all we need to formulate the equation of the line that we are looking for.

In the end, our answer will be x+3y=-3

Answer 2

Okay to find the perpendicular bisector of a segment you first need to find the slope of the reference segment.

m=(y2-y1)/(x2-x1) in this case:

m=(-5-1)/(2-4)

m=-6/-2

m=3

Now for the the bisector line to be perpendicular its slope must be the negative reciprocal of the reference segment, mathematically:

m1*m2=-1  in this case:

3m=-1

m=-1/3

So now we know that the slope is -1/3 we need to find the midpoint of the line segment that we are bisecting.  The midpoint is simply the average of the coordinates of the endpoints, mathematically:

mp=((x1+x2)/2, (y1+y2)/2), in this case:

mp=((4+2)/2, (1-5)/2)

mp=(6/2, -4/2)

mp=(3,-2)

So our bisector must pass through the midpoint, or (3,-2) and have a slope of -1/3 so we can say:

y=mx+b, where m=slope and b=y-intercept, and given what we know:

-2=(-1/3)3+b

-2=-3/3+b

-2=-1+b

-1=b

So now we have the complete equation of the perpendicular bisector...

y=-x/3-1 or more neatly in my opinion :P

y=(-x-3)/3