Answer:
- ∠ABC = 80°
- ∠ACB = 40°
- Extra distance: 122 km
Step-by-step explanation:
You want internal angles B and C, and the extra distance traveled to get from A to C via B.
a. Angles
The bearing of C from B is given as 320°. This is the angle measured clockwise from north. The same angle measured counterclockwise from north is 360° -320° = 40°.
The line BC is a transversal of the parallel north-south lines through B and C. Alternate interior angles at that transversal are congruent, so angle ACB will also be 40°.
Angle CAB is given as 60° (the bearing of B from A), so the triangle interior angle at B will be ...
∠ABC +∠ACB +∠CAB = 180°
∠ABC +40° +60° = 180° . . . . . . . use known angles
∠ABC = 80° . . . . . . . . . . . . . . . . subtract 100°
The angles of interest are ...
Distances
The sides of triangle ABC can be found using the law of sines.
[tex]\dfrac{AB}{\sin(C)}=\dfrac{BC}{\sin(A)}=\dfrac{AC}{\sin(B)}\\\\\\BC=AB\cdot\dfrac{\sin(A)}{\sin(C)}=(150\text{ km})\cdot\dfrac{\sin(60^\circ)}{\sin(40^\circ)}\approx202.09\text{ km}\\\\\\AC=AB\cdot\dfrac{\sin(B)}{\sin(C)}=(150\text{ km})\cdot\dfrac{\sin(80^\circ)}{\sin(40^\circ)}\approx229.81\text{ km}[/tex]
Flying to B caused the airplane to travel an additional ...
(150 km +202.09 km) -229.81 km = 122.28 km
The diversion caused the airplane to fly about 122 km farther.
