A line segment is bisected by a line with the equation - 2 y - x = 1 . If one end of the line segment is at ( 8 , 3 ), where is the other end?

1 Answer
May 31, 2016

-2y-x=1 bisects line segments between (8,3) and any point on the line -2y-x=16

Explanation:

Consider a horizontal line segment from (8,3) extending to the left
with a total length twice that of the horizontal distance from (8,3) to the line -2y-x=1.
From the diagram below we can see that this horizontal line will intersect -2y-x=1 at (-7,3)
and will have its left-most end at (-22,3)
enter image source here
Obviously this horizontal line is bisected by -2y-x=1

Consider a line through this left-most point with the same slope as -2y-x=1.
Using the slope-point form we can see that the equation of this line can be written as -2y-x=16

Assertion
for any point (x',y') on this new line, the line segment between (x',y') and (8,3) will be bisected by -2y-x=1

This Assertion follows immediately from considering a modified version of the above diagram.
enter image source here

Since QR||ST
color(white)("XXX")trianglePQR ~ trianglePST

color(white)("XXX")rArr PR:RT = PQ:QS = 1:1

rArr QR bisects PT for any point T=(x',y') on -2y-x=16