SOLUTION: Find the Equation of the angle bisector of the acute angles formed by the lines X+3y=9 and 4x+y=8.

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Question 1011464: Find the Equation of the angle bisector of the acute angles formed by the lines X+3y=9 and 4x+y=8.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The lines look like this:

For each line, all 3 coefficients are positive,
so the slope is negative and part of the line is in the first quadrant.
As we see in the drawing, the same will be true for the bisector of the acute angles.
If P%28x%2Cy%29 is a point in the angle bisector,
P is at equal distance from the lines forming the angle.
The distance from a point P%28x%2Cy%29 to a line ax%2Bby%2Bc=0 id
abs%28ax%2Bby%2Bc%29%2Fsqrt%28a%5E2%2Bb%5E2%29 ,
so for a point in the bisector of an angle formed by lines
x%2B3y=9<-->x%2B3y-9=0 and
4x%2By=8<-->4x%2By-8=0 ,
abs%28x%2B3y-9%29%2Fsqrt%281%5E2%2B3%5E2%29=abs%284x%2By-8%29%2Fsqrt%284%5E2%2B1%5E2%29
abs%28x%2B3y-9%29%2Fsqrt%2810%29=abs%284x%2By-8%29%2Fsqrt%2817%29
sqrt%2817%29%2Aabs%28x%2B3y-9%29=sqrt%2810%29%2Aabs%284x%2By-8%29 .
Two lines that are not parallel form, two angles, each with its own bisector.
In this case the bisectors are
sqrt%2817%29%28x%2B3y-9%29=sqrt%2810%29%2A%284x%2By-8%29<--> and
sqrt%2817%29%28x%2B3y-9%29=-sqrt%2810%29%2A%284x%2By-8%29<-->
The one with all positive coefficients is the bisector of the acute angles.
The bisector of the obtuse angles is perpendicular and has a positive slope (coefficients of x an y have opposite signs).