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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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+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

 

 

Triangle ADC:
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
avatar+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

 

 

cool cool cool

 Nov 18, 2019

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