# A line segment is bisected by a line with the equation # - 6 y + 2 x = 3 #. If one end of the line segment is at #( 4 , 8 )#, where is the other end?

##### 1 Answer

Any point on the line

#### Explanation:

Consider a horizontal line segment from

The equation of the horizontal line segment from

Noting that

The intersection of this horizontal line segment with the given bisector line will occur at

Continuing to travel horizontally a point twice as far away from

will be at

That is

For any point which could be such an endpoint, a line through this point parallel to the bisecting line will provide all possible bisected line segment endpoints.

Using the previously determined possible endpoint

we can write the slope-point version:

Any solution to the equation

Here is a graph of the point and the two lines in question:

graph{(-6y+2x-3)(3y-x+27)((x-4)^2+(y-8)^2-0.02)=0 [-25.65, 25.64, -12.83, 12.81]}