A line segment is bisected by a line with the equation # 9 y - 2 x = 5 #. If one end of the line segment is at #( 7 , 3 )#, where is the other end?

1 Answer
Dean R.
Jun 22, 2018

The general line containing all the endpoints of line segments with other endpoint #(7,3)# bisected by #9y-2x=5# is

#9y-2x=-3#

The endpoint of the perpendicular bisector is #(627/85, 111/85).#

Explanation:

If the line is the perpendicular bisector of the segment, the other endpoint is uniquely determined.

If the line is just a bisector, as asked, each point on the line is the bisector of some segment whose endpoint is #(7,3)# and whose other endpoint sweeps out a curve. Let's see if we can figure out its equation.

If we're told #(p,q)# is the midpoint of a segment with endpoint #(a,b)=(7,3)# the other endpoint #(x,y)# satisfies:

# (x,y)-(p,q)=(p,q)-(a,b)#

#(x,y)=(2p-a,2q-b)#

#(x,y)=(2p-7,2q-3)#

We have #(p,q)# on the line so #2p=9q-5#.

#(x,y)=(9q-12,2q-3)=(-12,-3)+q(9,2)#

The other endpoint #(x,y)# makes a line through #(-12,-3)# with direction vector #(9,2)# meaning slope #2/9.# Nonparametrically, that's the line

#y+3 = 2/9(x+12)#

#9y+27=2x+24#

#9y-2x=-3#

That's parallel to the original line, as far from it as #(7,3)# is.

Now let's ask which #(x,y)# makes a perpendicular bisector to the line?

The direction of the segment is #(x-a,y-b)=(x-7,y-3).# The direction of the line #-2x+9y=5# is #(9,2)# meaning for every #9# units #x# increases we increase #y# by #2# units.

We have perpendicularity when the dot product is zero:

#(x-7,y-3) cdot(9,2) = 0#

# 9(9q-12 - 7) + 2( 2q-3-3) = 0#

#85q - 183 = 0#

#q = 183/85#

#(x,y) = (9q-12,2q-3)=(627/85, 111/85)#

Check:

# ( (627/85,111/85)-(7,3) ) cdot (9,2) = 0 quad sqrt#

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