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

What we know:
line P endpoints (4,1) and (2,-5) (made up a line name for the this line)
perpendicular lines' slope are opposite in sign and reciprocals of each other
slope=m=(y2-y1)/(x2-x1)
slope intercept for is y=mx+b

What we need to find: 
line Q (made this name up for this line) , a perpendicular bisector of the line p with given endpoints of  (4,1) and (2,-5)

find slope of line P using (4,1) and (2,-5) 
m=(-5-1)/(2-4)=-6/-2=3

Line P has a slope of 3 that means Line Q has a slope of -1/3.

Now, since we are looking for a perpendicular bisector, I need to find the midpoint of line P to use to create line Q. I will use the midpoint formula using line P's endpoints (4,1) and (2,-5).

midpoint formula: [(x1+x2)/2, (y1+y2)/2)]
midpoint=[(4+2)/2, (1+-5)/2]
              =[6/2, -4/2]
              =(3, -2)

y=mx=b       when m=-1/3 slope of line Q and using point (3,-2) the midpoint                                 of line P where line Q will be a perpendicular bisector

(-2)=-1/3(3)+b               substitution
-2=-1+b                           simplified
-2+1=-1+1+b                 additive inverse
-1=b

Finally, we will use m=-1/3 slope of line Q and y-intercept=b=-1 of line Q
y=-1/3x-1