# A line segment is bisected by a line with the equation # 4 y + 3 x = 2 #. If one end of the line segment is at #( 2 , 1 )#, where is the other end?

##### 1 Answer

One possible other end would be at

The equation of all possible answers is

If the given equation is to be the **perpendicular** bisector:

#### Explanation:

Let

Since

and

The distance from

Moving another

i.e.

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**If we wanted the equation for all solution values**

Consider a line

Note the labeling of Line Segment

which we know from the previous work are in the ratio

Consider any other arbitrary Line Segment

Note again the labeling that divides

Since

the ratio of

That is any arbitrary line segment connecting

Since

its equation is

or

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**If we wanted #K# to be a #underline(color(black)("perpendicular"))# bisector of the line segment**

Consider the line

Since the slope of

the slope of

and the equation of

or

The endpoint of this perpendicular line segment can be derived as the intersection of

#{(4x-3y=5),(3x+4y=-6):}#

Which gives

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My apologies for the length of this solution.

If anyone can provide a complete solution more briefly, please post it.