The slope of a bisector

[um0123]

1. The problem statement, all variables and given/known data

A, B, and C, are points (1,2), (0,0), and (-1,3) respectively.

2. Relevant equations

Find the slope of the bisector of angle ABC.

3. The attempt at a solution

I have done the following and dont know how it helpsL

-found length of BA, BC, and AC
-found the slope of BA, and BC
-i dont know where to go from here

The length of BA is √5

The length of BC is √10

The length of AC is √5

The slope of BA is 2

The slope of BC is -3

Please do not do the work for me, i just need some advice as to what to do from here.

[um0123]

P.S. this is probably not precalculus, but this is problem that was in my precalculus book (i think we a re reviewing from algebra).

i forgot so much from last year:confused:

[Elucidus]

I'm not quite sure how'd one would go about doing this problem without a bit of trigonometry, but for starters, the segment AC has slope -1/2 and is therefore perpendicular to AB. Hence CAB is a right triangle (i.e. angle CAB is a right angle). This forces the distance of AC to be sqrt(15) (This is verifiable even if CAB isn't right).

But since CAB is a right triangle then the tangent of angle ABC is \sqrt(15)/\sqrt(5) =\sqrt(3). This along with a well timed tangent angle sum indentity should get you the answer.

--Elucidus