Find AB
Note that angle C = 180 - 45 - 15 = 120
Using the Law of Sines
12/ sin 45 = AB /sin 120
12 / ( 1/sqrt 2) = AB / (sqrt (3) / 2)
(12 sqrt 2) * (sqrt (3) / 2) = AB
12 sqrt 3 / sqrt 2 = AB
12 sqrt (2) sqrt (3) / 2 = AB
6 *sqrt 6 = AB
![[asy] unitsize(1.5 cm);pair A,B,C;A=(0,0);B=(5,0);C=(1,1);draw(A--B--C--A);label(](/api/ssl-img-proxy?src=https%3A%2F%2Flatex.artofproblemsolving.com%2Fc%2F5%2Fe%2Fc5e8916f6d09ff2ae02e073c9655077afad8d71c.png)
