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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.

 Jun 3, 2024
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    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

 

cool cool cool

 Jun 3, 2024

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