Question 1193847
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 What you have described is impossible.  With the given measurements, it is not possible for AD to bisect angle BAC.  That doesn't have anything to do with deriving the answers you need, but it is extraneous BS you shouldn't have included with the problem.


If BD measures *[tex \Large x] and DC measures 6, then BC must measure *[tex \Large x\,+\,6].  Then AB is the hypotenuse, so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ =\ (x\,+\,6)^2\ +\ 144]


So just simplify and solve the quadratic for *[tex \Large x] to get BD, and then calculate *[tex \Large 2x] to get AB.


Hint:  The quadratic factors.  Second hint: One of the roots is negative, so discard it because a negative length for a line segment is absurd.

																
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
*[illustration darwinfish.jpg]

From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> 
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