Answered By Eric C

Apply the sine law:

 $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$asinA =bsinB =csinC  

Refer to the drawing below (sorry for my bad drawing skills!):

The balloon, Patricia and Trent form 3 points of a triangle. We wish to know the length of the 'leg' connecting Trent and the balloon. This length is denoted by 'b'.

First, we must find the angle A, which is easy as we simply subtract the known angles from 180 degrees, giving us 106 degrees.

Substituting this into the equation:

 $\frac{a}{\sin A}=\frac{48m}{\sin106}=49.9m$asinA =48msin106 =49.9m 

Now by using the sine law:

 $\frac{a}{\sin A}=\frac{48m}{\sin106}=\frac{b}{\sin B}=\frac{b}{\sin43}$asinA =48msin106 =bsinB =bsin43  

b can be isolated: $b=a\cdot\frac{\sin B}{\sin A}=34.1m$b=a·sinBsinA =34.1m 

Do the same with c, the distance between Patricia and the balloon:

 $c=a\cdot\frac{\sin C}{\sin A}=25.7m$c=a·sinCsinA =25.7m 

Clearly, b is greater than c, therefore the distance between Trent (the person furthest away) and the balloon is 34.1 metres

Answered By Eric C

Sorry here's a slightly better picture