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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
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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
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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