Measuring the Distance between Two Points Remotely

 

Motivation:

                         Consider that the distance between two points is known,  we would like to find the distance between the other two points remotely based on the observation from two near points:

 

                                          

                         This question is of particular interest because we can not go to remote place to measure the distance over there.  Let’s think what kind of observation we need from point C and D in order to know the distance between A and B.

 

                         From Sine Law,  if the  two interior angles and the length of a side of a triangle are known,  we can know the lengths of the other sides.  From Cosine Law, if the lengths of two sides and the angle between them is known in a triangle,  we can know the length of the other side.

 

                                    

 

                         So, if the angles  1,  2,  3,  4 indicated above are also given, we can find the length of .  We are going to work on it in the following problem.

 

 

Question:     =d,   1= ,   2= ,  3= ,  4=  .  Find .

 

                                             

 

 

 

 

 

          Sol:

                            In  ACD,  we have

 

                                                     ( sine law )

                                    =

                                               =

 

                            Similarly in  BCD,  we have

 

                                             

                                   =

                                              =

 

                            Thus, in  ACB,  , , and  ACB are known.

                  can be obtained via Cosine law:

 

                        ² =  ² +  ² - 2  cos(  -  )

 

                So.