Tanker Bearings

Date: 06/11/2000 at 07:17:59
From: Rachel Crane
Subject: Bearings

I have my end of year exams coming up this week. One of the questions 
on the paper will be about bearings, and I don't understand a bit of 
it:

From a ship, A, the bearing of an oil tanker, T, is 300 degrees. 
From a second ship, B, which is 1000 m due west of A, the bearing of 
the oil tanker is 060 degrees. Explain why the oil tanker is the same 
distance from A as it is from B. (I also have to draw a rough sketch.)

Could you help me, please?

Thank you so much,
Rachel Crane

Date: 06/14/2000 at 08:30:18
From: Doctor Rick
Subject: Re: Bearings

Hi, Rachel,

A bearing is a direction, measured as an angle going clockwise from 
north. North is 0 degrees, east is 90 degrees, south is 180 degrees, 
and west is 270 degrees. In pure math the convention is to measure 
directions going counterclockwise from "east" (the positive x axis), 
but in practical fields like surveying or navigation, bearings are 
used. Don't ask me why there are two different systems; I guess the 
people who invented them just never talked with each other.

You can use bearings to draw the geometrical figures requested in your 
problem, and then use more familiar geometry to answer the question. 
Let's look at the problem:

>A) From a ship, A, the bearing of an oil tanker, T, is 300 degrees. 
>From a second ship, B, which is 1000 m due west of A, the bearing of 
>the oil tanker is 060 degrees. Explain why the oil tanker is the same 
>distance from A as it is from B. (I also have to draw a rough 
>sketch.)

Draw a point A. Then draw a ray from A at a bearing of 300 degrees. 
Going 300 degrees clockwise from north (up, in your drawing) is the 
same as going 60 degrees counterclockwise, because 300 + 60 = 360 
degrees, a full circle. We don't know where P is yet, so I'll label a 
point on the ray Q just so we can identify the ray. It will look like 
this:

     Q _           North
      |\           ^
          \        |
             \   60|
                \  |
                   *A

Draw a line from A to B. It goes west (to the right) from A. Make it 
any length but label it 1000 m.

     Q _           N
      |\           ^
          \        |
             \   60|
             30 \  |
     B*------------*A

What is the angle between AQ and AB? Since angle NAB is a right angle, 
angle NAQ + angle QAB = 90 degrees. Therefore angle QAB = 30 degrees.

We have one more ray to draw, the ray BP. Its bearing is 060, that is, 
60 degrees clockwise from north. I'll let you draw it because my 
figure would get too complicated. Do the same thing I did above: draw 
a line north from B, and mark off an angle of 60 degrees on the right 
side of that line. What is the angle between the ray you have drawn 
and BA?

Mark P on the figure, at the intersection of AQ and the new ray. Now 
you have enough information to answer the question.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 06/15/2000 at 11:38:43
From: Rachel Crane
Subject: Re: Bearings

Thanks for your message, Dr. Rick. It has helped me a lot. All I need 
now is some good luck for tomorrow's math test!

Cheers,
Rachel Crane