Law of Sines

**alexthesauceboss** · January 21st 2013, 12:54 PM

The bearing from the Pine Knob fire tower to the Colt Station fire tower is N 65 degrees East, and the two towers are 30 km apart. A fire spotted by rangers in each tower has a bearing of N 80 degrees East from Pine Knob and S 70 degrees E from Colt Station. Find the distance of the fire from each tower.

**chiro** · January 21st 2013, 06:29 PM

Hey alexthesauceboss.

Are the two triangles right angled ones?

**Soroban** · January 21st 2013, 08:35 PM

Hello, alexthesauceboss!

$\text{The bearing from the Pine Knob fire tower to the Colt Station fire tower is }N\,65^o\,E.$
$\text{The two towers are 30 km apart. }\,\text{A fire spotted by rangers in each tower.}$
$\text{It has a bearing of }N\,80^o\,E\text{ from Pine Knob and }S\,70^o\,E\text{ from Colt Station.}$
$\text{Find the distance of the fire from each tower.}$

The diagram looks like this:

Code:

                          C
                          o
                        * : *
      N               *65^o:70^o*
      :             *     :     *
      :        30 *       :       *
      :         *         :     30^o o F
      :       *           :   *
      : 65^o *           * :
      :   * 15^o   *       S
      : *   *
    P o

Pine Knob is at $P$ , Colt Station is at $C$ .
$\angle N\!PC = 65^o,\;PC = 30.$

The fire is at $F.$
$\angle N\!PF = 80^o \quad\Rightarrow\quad \angle CPF = 15^o.$
$\angle FCS = 70^o \quad\Rightarrow\quad \angle PCF = 135^o \quad\Rightarrow\quad \angle F = 30^o$

Law of Sines:

. . $\frac{CF}{\sin15^o} \:=\:\frac{30}{\sin30^o}$

. . $\frac{PF}{\sin135^o} \:=\:\frac{30}{\sin30^o}$

Go for it!

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