+118 In triangle ABC, point D is on side AC such that line segment BD bisects angle ABC. If angle A = 30, angle C = 90, and AC = 12, then find the area of triangle ABD.
+129852 B
30 30
90C D A30
12
BC = 12/sqrt 3 = 4sqrt 3
BA = 24/sqrt 3 = 8 sqrt 3
Since BD is a bisector
BC / BA = DC / DA
[4sqrt 3]/ [8/sqrt 3] = DC /DA
4/8 = DC/ DA
1/2 = DC / DA
Therefore DA = 2 / [1 + 2 ] AC = (2/3)AC = (2/3) 12 = 8
[ABD] = (1/2) (BA) ( DA) sin (30) = (1/2) (8sqrt 3)(8) (1/2) = 16 sqrt 3
