### Video Transcript

In this video, we’ll learn how to
draw and measure bearings. We begin, though, by actually
discussing what a bearing is and why they’re seemingly so important in the world of
transport.

Imagine you’re on a ship in the
middle of the ocean and you’re looking to navigate to an island. Unlike if you were in a car, you
cannot simply look for points of reference, like an oak tree or a supermarket or
even road names and navigate using those. So, instead, we use something
called a bearing. A bearing is a fancy way of
measuring an angle. But, of course, we need to ensure
that every ship captain and indeed every pilot of a plane measures these angles in
the same way. And so, we apply some rules.

The rules for measuring and drawing
bearings are as follows. First, we measure from north or the
north line. Now, this means that if there isn’t
a north line drawn on the diagram, we do need to add one. Next, we measure from our north
line in a clockwise direction. Finally, we tend to use three
digits to describe our bearings. So, 43 degrees as a bearing would
be zero, four, three. Now, obviously, this is a great
idea in navigation because if you’re communicating with someone else, you’ll know
that you haven’t missed any digits if you use three figures all the time.

Depending on where you are in the
world, though, when answering questions that don’t involve transport, you might find
that 43 degrees would simply be written as 43 degrees. In some places, this would even be
written using compass points such as north 43 west, and so on. The terminology also takes a little
bit of getting used to. For example, if we’re measuring a
bearing of 𝐵 from 𝐴, we’re going to measure the bearing at 𝐴. And so in the case of our diagram,
we start at the north line, measure the angle in a clockwise direction that this
north line makes with the line segment 𝐴𝐵.

Finally, it’s worth noting that we
can use bearings to describe specific compass points. We start at north and travel in a
clockwise direction. So, east is a bearing of 090. South would be a bearing of 180
degrees. West sits on a bearing of 270. And the north line itself
represents a full turn; it’s a bearing of 360. We’ll now look at how we can use
these facts to measure very simple bearings.

Find the bearing of 𝐵 from 𝐴.

Let’s begin by recalling what we
understand by the word “bearing.” A bearing is a fancy way of
measuring an angle. When we measure bearings, we
remember three things. We first measure from north or from
a north line. Once we’ve identified that north
line, we measure in a clockwise direction. In navigation, we also use three
digits to describe our bearing. Although depending on where you are
in the world, when we’re not working with navigation problems, we’ll sometimes use
two digits to represent a two-digit number or even include compass points.

The question wants us to find the
bearing of 𝐵 from 𝐴, so we’re going to measure the bearing at 𝐴 using the rules
we’ve given. Firstly, we identify the north line
at 𝐴. We then travel in a clockwise
direction until we hit the line segment that joins 𝐴 to 𝐵. According to our diagram, the angle
that these two lines make with one another is 75 degrees. And so, if we were going to be
following rule three, we would represent the bearing of 𝐵 from 𝐴 as 075. Now, of course, since this isn’t a
navigation problem, the use of three digits isn’t quite as necessary. So, this might also be written as
75 degrees. The bearing of 𝐵 from 𝐴 then is
075 or 75 degrees.

In our next example, we’ll look at
how to find the location of an object, given information about bearings from two
different points.

The diagram shows the position of
two lighthouses, A and B. A ship is on a bearing of 068
degrees from lighthouse A and a bearing of 295 degrees from lighthouse B. Mark the position of the ship on
the diagram.

We’re given the location of this
ship relative to two points, A and B. In fact, we’re told the location in
terms of its bearings. So, let’s recall what we mean by a
bearing. A bearing is a fancy way of
measuring an angle, and we remember that we begin by measuring from the north
line. We always measure in a clockwise
direction from this north line round to the line segment we’re interested in. And, when necessary, when working
with navigation, we try to use three digits or three-figure bearings. And so, a bearing of 068 is an
angle of 68 degrees from the north line. So, we’ll begin by measuring this
bearing, a bearing of 68 or 068 degrees from lighthouse A.

We first locate the north line at
A. Because we’re looking to measure in
a clockwise direction, we place the protractor as shown. By traveling up the north line, we
see that the zero we’re looking for is in the outer row of numbers. And so, we travel in a clockwise
direction around this outer row of numbers until we find an angle of 68 degrees. 68 degrees on our outer row of
numbers is here. So next, we’re going to remove the
protractor and join A to this point. We draw a single line segment from
A through this point, as shown.

We’re now going to repeat this
process for our second bearing; that’s a bearing of 295 degrees from lighthouse
B. Once again, we begin by locating
the north line. We’re now going to go in a
clockwise direction from this north line. Note that we’re looking to measure
a bearing of 295 degrees. And so, we have a little bit of a
problem. If we were to measure in a
clockwise direction and place our protractor as shown, we see that, in this
direction, we can only go as far as 180 degrees. And so, we mark 180 degrees on our
diagram and we rotate our protractor. Our job now is to figure out
exactly how much further around we need to go to hit a bearing of 295 degrees.

Now what we know is that we’ve
already traveled 180 degrees in a clockwise direction. So, let’s subtract 180 from
295. 295 minus 180 is 115. And so, we’re going to travel
around. Once again, we’re using the outer
row of numbers because we’re starting at zero, and we’re going to go in a clockwise
direction round to 115. 115 degrees is here. Let’s remove the protractor and
join this up with point B. Let’s draw a nice long line segment
from B and through this point, as shown. We know that the ship sits on a
bearing of 068 degrees from lighthouse A and 295 degrees from lighthouse B. This must mean the ship is located
where our two lines representing these bearings overlap. That’s here. And so, we’ve marked with a cross
the location of the ship on our diagram.

Note that there was another way we
could’ve calculated the exact location of our second bearing. We know the angles around a point
sum to 360 degrees. So, we could alternatively have
subtracted 295 from 360 and then measured 065 or 65 degrees in a counterclockwise
direction.

We’ll now look at something called
back bearings or reverse bearings.

Find the bearing of 𝐴 from 𝐵.

We first recall that a bearing is
simply a way of measuring an angle. We remember three things. We measure from the north line, and
we always do so in a clockwise direction until we reach our line segment. In navigation, we also use three
digits. So, for example, 73 degrees as a
bearing would be 073. Now, the question here wants us to
find the bearing of 𝐴 from 𝐵. This means we’re going to be
measuring the bearing at 𝐵. So, let’s add a north line in. Now, since this diagram may not
necessarily be to scale, we’re going to use some rules for working with parallel
lines to deduce the bearing of 𝐴 from 𝐵.

Remember, we measure in a clockwise
direction from our north line round to the line segment that joins 𝐴 to 𝐵. So, that’s this angle shown. And so, to calculate this angle,
we’re going to carry the north line a little bit down from 𝐵. We know that our north lines must
be parallel and that alternate angles are equal, so we can mark this angle on our
diagram as being equal to the angle given; it’s 115 degrees. We also know that angles on a
straight line sum to 180 degrees. So, this angle that I’ve marked
along our north line is 180. It follows then that the bearing of
𝐴 from 𝐵 must be the sum of these two angles. It must be 180 plus 115. 180 plus 115 is 295. And so, we see the bearing of 𝐴
from 𝐵 is 295 degrees.

This is sometimes called a back
bearing or a reverse bearing, and we can generalize this a little. We say that the difference between
a bearing and its reverse bearing will always be 180 degrees. And so, given a bearing of 𝐴 from
𝐵, we find the bearing of 𝐵 from 𝐴 by either adding or subtracting 180
degrees. Whether we add or subtract will, of
course, depend on the size of the original bearing. In this case, for example, we
wouldn’t subtract 180 since 115 minus 180 is negative, and we know that we don’t
work with negative bearings.

In our next example, we’ll look at
how we can combine bearings with points on the compass.

A plane flies on the bearing
shown. The control tower told the pilot to
fly due west toward the airport. Which of the following is the angle
the pilot should turn through? Is it (A) 137 degrees clockwise,
(B) 90 degrees counterclockwise? Is it (C) 270 degrees
counterclockwise, (D) 133 degrees counterclockwise, or (E) 223 degrees
clockwise?

We’re given a path of a plane in
our diagram and we’re told that, at some point, the control tower tells the pilot to
fly due west. We know that relative to the north
line, to travel west, we’d move to the left on our diagram. And so, we see that the plane is
going to travel left on our diagram. Now, given the direction the plane
is traveling, there are two ways the pilot could achieve this. Firstly, they could travel in a
counterclockwise direction as shown. A slightly longer route will be to
travel in a clockwise direction. Let’s begin by looking at the
counterclockwise direction since this is slightly shorter.

Now, there are a number of ways we
can find this angle. One way is to add a north line at
the location of the pilot. We then know the two north lines in
our diagram are parallel. So, we can add 43 degrees here,
since we know that corresponding angles are equal. We also know that the north line
and the west line are perpendicular. They meet at an angle of 90
degrees. And so, we can calculate the angle
that the pilot turns through by adding 90 and 43. 90 plus 43 is 133 degrees. And so, the pilot could turn 133
degrees counterclockwise. And that’s option (D).

But remember, we said that the
pilot could have turned in the opposite direction. They could have traveled in a
clockwise direction. So, how could we have calculated
this angle? Well, by extending the west line
and recalling that the north line and west line are perpendicular, we can draw a
right-angle triangle, as shown. We know that angles in a triangle
sum to 180 degrees. So, we can find the third angle in
this right-angle triangle by subtracting 90 and 43 from 180 to get 47 degrees.

We know that angles on a straight
line sum to 180 degrees. And so, had the pilot turned on a
clockwise direction, the angle would’ve been calculated by adding 180 to 47 to get
227 degrees. And so, this isn’t one of our
options. But we could have said that the
pilot needed to turn on an angle of 227 degrees clockwise.

In our final example, we’ll look at
how we can use a little bit of geometry to solve bearings problems.

Two roads 𝑂𝐴 and 𝑂𝐵 are shown
in the diagram. The roads meet at an angle of 112
degrees. Jackie is at point 𝑂, the
intersection of the two roads. She walks towards 𝐴. On what bearing does she walk?

She is currently at point 𝑂, and
she wants to walk towards point 𝐴. And so, we’re going to need to
calculate the bearing of 𝐴 from 𝑂. We recall that when we measure and
draw bearings, we begin by looking at the north line. We then measure in a clockwise
direction until we hit the line segment we’re interested in. So that’s the line segment
𝑂𝐴. So, how are we going to calculate
this angle? We know that angles on a straight
line add to 180 degrees. And so, we can find the measure of
the angle we’re looking for by subtracting the given angle, 112 degrees, from
180. 180 minus 112 is 68. The angle that we’re looking for
then is 68 degrees.

And remember, when we’re working,
particularly, with navigation problems, we tend to use three-figure bearings. So, as a bearing, 68 degrees is
068. Jackie must walk on a bearing of
068 degrees towards 𝐴.

In this video, we saw that a
bearing is a fancy way of measuring an angle, and it’s often used in navigation. We saw that to measure and draw
bearings, we begin by identifying the north line. Once we’ve identified that north
line, we measure our angle in a clockwise direction round to the line segment we’re
interested in. And that we tend to use three
digits to represent our bearings. We have problems that don’t involve
any navigation. Every so often, this might just be
simply represented using a two-digit number or, alternatively, by referencing the
compass points.

Finally, we saw that the difference
between a bearing and its reverse bearing or back bearing is always 180 degrees. So, given a bearing of 𝐴 from 𝐵,
we calculate the bearing of 𝐵 from 𝐴 by either adding or subtracting 180 degrees
and making sure that we don’t get a negative value.