Last modified 30 th December 2013 A locus is a set of points satisfying a certain condition Loci Thing A Thing B Loci involving Interpretation A given distance from point A Point Resulting Locus ID: 816121
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Slide1
Year 9: Loci
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Last modified: 30
th
December 2013
Slide2!
A locus is a set of points satisfying a certain condition.
Loci
Thing A
Thing B
Loci involving:
Interpretation
A given distance from point A
Point
Resulting Locus
-
Click to Learn
A
A given distance from line A
Line
-
Click to Learn
A
Equidistant from 2 points or given distance from each point.
Point
Point
Click to Learn
A
B
Perpendicular bisector
Equidistant from 2 lines
Line
Line
Click to Learn
A
B
Angle bisector
Equidistant from point A and line B
Point
Line
Not until FP1 at Further Maths!
B
Parabola
A
?
?
?
?
?
Slide3Fixed distance from a point
A goat is attached to a post, by a rope of length 3m. Shade the locus representing the points the goat can reach.
3m
Click to
Broshade
Moo!
Slide4Fixed distance from a point
A goat is now attached to a metal bar, by a rope of length 3m. The rope is attached to the bar by a ring, which is allowed to move freely along the bar.
Shade the locus representing the points the goat can reach.
3m
Click to
Broshade
Common schoolboy error: Thinking the locus will be oval in shape.
Slide5I’m 2m away from the walls of a building. Where could I be?
Copy the diagram (to scale) and draw the locus. Ensure you use a compass.
Circular corners.
Straight corners.
10m
Scale: 1m
:
1cm
2m
2m
2m
10m
Exercise
Q1
Slide610m
Scale: 1m
:
1cm
2m
10m
Exercise
I’m 2m away from the walls of a building.
Copy the diagram (to scale) and draw the locus. Ensure you use a compass.
Q2
6m
6m
Click to
Broshade
Slide7My goat is attached to a fixed point A on a square building, of
5m x 5m, by a piece of rope 10m in length. Both the goat and rope are fire resistant. What region can he reach?
5m
10m
A
Exercise
Q3
Scale: 1m
:
1cm
Bonus question:
What is the area of this region, is in terms of
?
87.5
?
Click to
Broshade
Slide8Distances from two points
Maxi is phoning his friend to get a lift to a party. He says he is 3km away from Town A and 5km from Town B.
Sketch the locus his friend needs to check to find Maxi.
3km
Click to
Brosketch
A
B
5km
Bonus Question: How could Maxi augment his description so the locus is just a single point?
He just needs a third landmark to describe his distance from.
The process of determining location using distances from points is known as
trilateration
, and is used for example in GPS. It is often confused with
triangulation
, which uses
angles
to determine location rather than distances.
?
Q4
Slide9A
B
3km
4km
A goat is at most 3km from A and at least 4km from B.
Shade the resulting locus representing the region the goat can be in.
Distances from two points
Q4
Click to
Broshade
Slide10Equidistant from 2 points
But now suppose we don’t have a fixed distance from each point, but just require the distance from both points to be the same. What is the locus now?
A
B
STEP 1:
Put your compass on A and set the distance so that it’s slightly more than halfway between A and B. Draw an arc.
STEP 2:
Using the same distance on your compass, draw another arc, ensuring you include the points of intersection with the other arc.
STEP 3:
Your locus is the line that goes between these points of intersection.
It is known as the
perpendicular bisector
.
Slide11Common Losses of Exam Marks
A
B
Le
Problemo
:
Arcs don’t overlap enough, so points of intersection to draw line through is not clear.
A
B
Le
Problemo
:
Locus is not long enough.
(Since it’s actually infinitely long, we want to draw it sufficiently long to suggest it’s infinite)
?
?
Slide12Equidistant from two lines
STEP 1:
Measure out some distance across each line, ensuring the distance is the same.
STEP 2:
The locus is just the perpendicular bisector of these two points.
The line is known as the
angle bisector
because it splits the angle in half.
A
B