The slope of a bisector

1. Aug 31, 2009

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.

2. Aug 31, 2009

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

3. Aug 31, 2009

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