Mr Barton’s Maths Notes
Shape and Space
4. Loci
www.mrbartonmaths.com
With thanks to www.whiteboardmaths.com for the images!
4. Loci
What on earth is Loci?
• Loci is all about tracing the paths of points as they move following certain rules
• It has many real-life applications, especially for architects and builders who want to make sure
things go in the right place and they don’t run out of room
• Note: If you are one of those people who doesn’t like the number and algebra bits of maths,
then this could be the very topic for you!
What we are going to do in this section
• Instead of going through how to do things like draw angle-bisectors, I am going to pick out a
few of the classic type of Loci questions I have seen come up in exams in the past and take you
through, step-by-step, how to do each one.
NOTE: It is probably worth while reading through 8. Constructions before carrying on, as some
of the skills you need are explained in greater detail there!
Example 1
My pet penguin has been tied up by a 10 metre rope to the corner of the shed as
shown below. Draw and shade the area which my penguin can move
Skills needed: drawing circles with compass
Scale:1cm = 2m
Shed
Steps:
1. Firstly, we need to sort out our scale – every 1cm square is equal to 2metres in real
life – so the 10m rope our penguin is tied to is in fact… 5cms long!
2. Now, we want to see how far our penguin can go in all directions. So, we must draw
a circle with our compass (radius 5cm) and with the centre at the point on the
shed where the penguin is tied.
Watch Out! But that’s not the full story… because walls of the shed prevent the
penguin from going quite as far upwards – he cannot walk through walls!
He can go along the side of the shed to point B, which is 3cms away, and once he has
reached this point, he can go another 2cms in any direction.
3. So… we must now set our compass again and draw a circle with radius 2cm and
centre at point B.
4. We now have the area where the penguin can walk, so we can shade it in!
Scale:1cm = 2m
B
Shed
Example 2
A farmer wants to lay a water pipe across his field so that it is equidistant from two
hedges. He also wants to connect a sprinkler in the exact centre of the pipe, that waters
the field for 40 metres in all directions.
Skills needed: bisecting angles and bisecting lines
(a) Show the position of the pipe
inside the field.
C
D
(b) Mark the point of connection
for the sprinkler.
(c) Show the area of the field that
is watered by the sprinkler.
B
E
A
Scale:1cm = 20m
(a) Show the position of the pipe
inside the field.
C
Steps:
D
1. Firstly, we need to realise what
the question is asking… the pipe
must always be the same
distance from line AB as line
AE… well, the only way to do
that is to bisect the angle at A!
2. Place the pointy bit of your
compass at A and mark a point
on AE and AB
B
E
A
3. Now place your pointy bit on
each of these new points and
draw two arcs in the centre of
the shape
4. Mark a new point where these
two arcs cross
5. Draw a line that starts at A and
goes through this crossing point
and voila!... There is your pipe!
(b) Mark the point of connection
for the sprinkler.
Steps:
C
D
B
E
A
1. Okay, so we have to find the
exact centre of the pipe. Now
it might be tempting to try to
do it with your ruler… but that’s
no fun, and more importantly,
it’s not accurate! Instead, we
must bisect the line
2. Place the pointy bit of your
compass at A draw an arc on
the right and an arc on the left
3. Place the pointy bit of the
compass at the other end of
the pipe and do the same.
4. Mark two points where these
arcs cross
5. Draw a line through the two
crossing points and where it
hits the pipe is the exact
centre!
(c) Show the area of the field that
is watered by the sprinkler.
C
Steps:
D
1. First we must check our scale…
1cm = 20m, and we want to
water 40m… so that is 2cm on
our drawing!
2. The water can travel 2cm in all
directions, so we must draw a
circle
B
E
A
Scale:1cm = 20m
3. Place the pointy bit of the
compass at the centre of the
pipe and draw a circle with
radius 2cm
4. Shade in the circle and you are
done!
Good luck with
your revision!