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  # help
  
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  Given that AB = 7.7, angle CAB = 36 degrees, BC = 9, and angle ACD = angle BCD.  Find AD.
  
  Nov 18, 2019
  
  #1
  +2
  
  Triangle ABC:
  Using Law of Cosines:
  Sides: AB = 7.7 BC = 9 AC = 14.009
  Angle ∠ A = 30.191° = 30°11'27″ = 0.527 rad
  Angle ∠ B =  36° = 0.628 rad
  Angle ∠ C = 113.809° = 113°48'33″ = 1.986 rad
  
  Using Law of Sines:
  Sides: AD = 4.688 DC= 10.581 AC= 14.009
  Angle ∠ C = 15.096° = 15°5'46″ = 0.263 rad
  Angle ∠ A = 36° = 0.628 rad
  Angle ∠ D = 128.904° = 128°54'14″ = 2.25 rad
  
  Nov 18, 2019
  #2
  +106515
  +1
  
  Using the Law of Sines
  
  sin 36 / 9  =  sin ACB / 7.7
  
  sin ACB = 7.7 sin 36 / 9
  
  arcsin (7.7 sin 36 / 9)  = ACB  ≈  30.19°
  
  So angle  CBA = 180 - 36 - 30.19  ≈  113.81°
  
  So....by the Law of sines
  
  AC/ sin 113.81 = AB/ sin 30.19
  
  AC / sin 113.81  =  7.7 / sin 30,19
  
  AC = 7.7 sin13.81 / sin 30.19   ≈  14.0089
  
  And because the apex angle is bisected
  
  AC / AD  = BC / DB
  
  AC / BC  =  AD/ DB
  
  14.0089 / 9  = AD / DB
  
  ( 14.0089 / 9) DB = AD
  
  (14.0089 / 9) DB + DB  = 7.7
  
  (1 + 14.0089/9) DB = 7.7
  
  DB =  7.7 / ( 1 + 14.0089/9) ≈   3.011
  
  So
  
  AD  = 7.7 - 3.011  ≈ 4.689
  
  Nov 18, 2019