Given that AB = 7.7, angle CAB = 36 degrees, BC = 9, and angle ACD = angle BCD. Find AD.
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
+128460 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
