Hello, Tweety!

Town B is 6 km, on a bearing of 020°, from town A.

Town C is located on a bearing of 055° from town A and on a bearing of 120° from town B.

Work out the distance of town C from (a) town A and (b) town B.

Bearings are measured clockwise from North. Code:

Q
:
:
B o 120°
P /: *
| / :60°*
| / : *
| /20°: *
| / : 70° o C
|20°/ : *
| / : *
| / 30° * :
|/ * :
A o R

$\displaystyle \angle PAB \:=\: 20^o\:=\:\angle ABR,\:AB = 6\text{ km}$

$\displaystyle \angle PAC \:=\:50^o \quad\Rightarrow\quad \angle BAC \:=\:30^o$

$\displaystyle \angle QBC \:=\:120^o \quad\Rightarrow\quad \angle CBR \:=\:60^o\quad\Rightarrow\quad \angle ABC \:=\:80^o$

In $\displaystyle \Delta ABC\!:\;\;\frac{AC}{\sin80^o} \:=\:\frac{6}{\sin70^o} \quad\Rightarrow\quad AC \:=\:\frac{6\sin80^o}{\sin70^o} \:\approx\:6.3\text{ km}$

. . . . . . . . $\displaystyle \frac{BC}{\sin30^o} \:=\:\frac{6}{\sin70^o} \quad\Rightarrow\quad BC \:=\:\frac{6\sin30^o}{\sin70^o} \:\approx\:3.2\text{ km}$