Geometry - please explain your answer without the use of cosines

By sine law,

AB = AD*sin(angle BAD)/sin(ABD) = BD/sin(BCD)*sin(BDC) = 30/72*24 = 10

In right triangle DBC, the sides are: BC==78, BD==30,  DC==72. Angle BDC==90, Angle DBC==67.38, Angle BCD ==22.62 degrees

 

In right triangle ABD, the angles are: BDA==90,  DBA==22.62,  DAB==67.38

 

Sine(67.38) ==30 / AB

0.923076 ==30 / AB

AB ==30 / 0.923076

AB ==32.50