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Question 1086403: what is the equation of the line bisector of the acute angle formed by the intersection of the lines 4x+3y-24=0 and 5x-12y+30=0
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  shows  and  lines.
Those lines form four angles.
Because those line are are not perpendicular to each other,
two of those angles are acute and two are obtuse.
The bisector of an angle is the locus of the points that are
at equal distance from the two sides of the angle.
Distance from a point  to a line  is
 .
So, the lines bisecting the angles formed by  and 
have the equations
   .
That simplifies to
  ,
and to
 
and
  .
The plus sign would give us
 <-->  <-->  <-->  ,
with a slope of   ,
showing a slight downward slope.
The minus sign would give us
 <-->  <-->  ,
with a slope of   ,
showing a very steep upwards slope.
The line has a shallow upwards slope of  .
The line has a slope of  (downwards).
A line perpendicular to would have a steeper upwards slope of
 .
So, the acute angle between and
includes shallower slopes between  and  ,
such as  , not steeper slopes, such as  or .
That means that the acute angle bisector line is
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