LOCI

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LOCUS POINTS
LOCI

Studying the path of moving points and using them
to make accurate drawings
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LOCI deals with the path drawn out by a moving
point. The LOCUS of a point will show all the
possible positions of that point as it follows a
certain rule.

Lets have a look at an example


Imagine a point on a piece of paper.


Now draw another point which is 3cm away.
Draw some more points which are all 3cm away from
the original.






Now lets put a few more on there.

Now lets shrink the crosses to small points and
show all of the possible points which are 3cm
away from the original.


The green part shows all the possible positions
of points which lie 3cm away from the original
point. As you can see, they form a circle.
WHAT IS THE RADIUS OF THIS CIRCLE?
Answer 3cm
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LOCI deals with the path drawn out by a moving
point. The LOCUS of a point will show all the
possible positions of that point as it follows a
certain rule.
Lets have a look at another example
Look at the line below and imagine all of the
points which lie 2cm from it.
Lets start to draw them in green.
Which points have been missed out?
Dont forget the ones on the other side of the
line too.
The green part shows all the possible positions
of points which lie 2cm away from the original
line. As you can see, they form two parallel
lines which lie 2cm at either side of the
original line.
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LOCI deals with the path drawn out by a moving
point. The LOCUS of a point will show all the
possible positions of that point as it follows a
certain rule.
Lets see what happens at the ends of a line.
Lets draw all of the points which lie 2cm from
this line.
Whilst the line is straight, its just like the
last example. A pair of parallel lines at either
side of the original.
What do you think happens at the ends of the line?
Lets have a look to see if you were correct.
The green part shows all the possible positions
of points which lie 2cm away from the original
line. The resulting shape looks similar to a
running track
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OK! Now its time for you to try one.
Firstly draw an L shape with these dimensions.
We are carefully going to draw all of the points
which lie 2cm from this L shape.
Before starting, try to imagine what it will look
like when its finished.
Lets start by drawing in those parallel lines,
but be very careful on the interior corner.
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Now we can deal with the ends of the lines. Use a
compass set at 2cm for these
Now lets deal with the corner.
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The green part shows all the possible positions
of points which lie 2cm away from the original L
shape.
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Now try these ones on your own.
Draw each of the following shapes out using the
dimensions shown and then mark on the lines which
represent all the points which follow the given
rule.
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How well did you do?
2
1
3
3
3
3
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Exam Style Question
At a zoo, tigers are kept in a large enclosure. A
tiger can reach through the bars of the
enclosure and use its claws to grab anything up
to a metre away. The zoo keepers want to erect a
fence to make sure that visitors to the zoo do
not stroll into the reach of the tigers. Draw
out the enclosure from this diagram and show
where they should erect their fence around this
part of the cage.
3m
Lets see how you did..
2m
2m
3m
2m
2m
6m
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Using Loci
Bobs house lies exactly 200m from the river and
exactly 300m from the church. Use the position of
the river and the church to find possible
locations for Bobs house.
Church

Here are the points that are 300m from the church.
So these are the only two possible positions of
Bobs house.
River
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A Practical Example
A thief steals a helicopter and takes off from
Bolton with 40 miles worth of fuel onboard. The
police use a map to show where the thief could
land the helicopter.
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An Interesting Example
Pick a point on the outside of a wheel and look
at the path that it takes as the wheel rolls
along the ground. The animation below traces out
the locus of the point as the wheel turns.