Loci
Objectives:
Grade D
Understand the idea of a locus
Grade C
Construct accurately loci, such as those of points
equidistant from two fixed points
Solve loci problems, such as identifying points
less than 3cm from a point P.
Prior knowledge:
Construct the perpendicular bisector of a line
Construct the bisector of an angle
Loci
A Locus is the path followed by a moving point
Loci is the plural of locus
Loci can come in various forms:
1.
The locus of points from a point P ....
P
5 cm
x
... is a circle radius 5 cm, centre P
Loci
2. The locus of points the same distance from line AB
The locus of points 2 cm from line AB
A
B
2 cm
Loci
3. The locus of points equidistant from two fixed points
equidistant means:
A and B
‘same distance from’
A
B
locus of points equidistant from A and B
(perpendicular bisector of AB)
Loci
4. The locus of points equidistant from two fixed lines AB and AC
B
A
C
locus of points equidistant from AB and AC
(angle bisector of angle BAC)
Important Note
Loci
For a rectangular shape a locus inside
is different from a locus outside
Inside
Sharp corners
Outside
Rounded corners
Worked Example 1
Loci
A dog is on a 2 metre lead which is tethered to a 5m long pole in
the centre of a garden. The garden has a flower border of ½ m
inside the garden wall. Show the loci of these on the diagram below.
Flower border
Scale: 1 large square = 1 metre
Loci
Worked Example 2
A treasure chest is buried on an island. The map of the island is below.
The treasure is equidistant from A and B and also equidistant from
C and D. Mark the position of the treasure on the map.
B
C
X
A
locus of points
equidistant
from
C and D
D
locus of points equidistant from A and B
Loci
Question 1
8
6
8 cm
6 cm
Loci
Question 2
•
•
•
Mark 2 points A and B, which are 10 cm apart
Construct the locus of all the points that are exactly 6 cm from B
Shade in all the points that are less than 6 cm from B and closer
to A than B
A
B
Question 3
Loci
A rectangular lawn is 12 m long and 10 m wide.
The gardener has three sprinklers, which spray water in a circle of radius 3m.
Sprinkler A is 3m from the top edge and 3m from the left edge.
Sprinkler B is 3m from the top edge and 3m from the right edge.
Sprinkler C is 3m from the bottom edge and 6m from the left edge.
Make a scale drawing of
the garden and shade
the area that is covered by
more than one of the
sprinklers
A
B
C
Loci
Question 4
The shape ABCD below represents a field on a scale of 1 cm to 10 m.
Construct a line AE such that it bisects angle BAD and E is a point on the line BC
Construct the locus of points that are 40m from E
Construct the
locus around E
4cm (40m)
E
B
C
Construct the
angle bisector
A
D
Shade the area of the field that is more than 40 m from E and closer to AB than AD
Worked Example 1
Loci
A dog is on a 2 metre lead which is tethered to a 5m long pole in
the centre of a garden. The garden has a flower border of ½ m
inside the garden wall. Show the loci of these on the diagram below.
Scale: 1 large square = 1 metre
Loci
Worked Example 2
A treasure chest is buried on an island. The map of the island is below.
The treasure is equidistant from A and B and also equidistant from
C and D. Mark the position of the treasure on the map.
B
C
A
D
Loci
Question 1
8
6
Question 2
•
•
•
Loci
Mark 2 points A and B, which are 10 cm apart
Construct the locus of all the points that are exactly 6 cm from B
Shade in all the points that are less than 6 cm from B and closer
to A than B
Question 3
Loci
A rectangular lawn is 12 m long and 10 m wide.
The gardener has three sprinklers, which spray water in a circle of radius 3m.
Sprinkler A is 3m from the top edge and 3m from the left edge.
Sprinkler B is 3m from the top edge and 3m from the right edge.
Sprinkler C is 3m from the bottom edge and 6m from the left edge.
Draw the garden and shade
the area covered by the sprinklers
Loci
Question 4
The shape ABCD below represents a field on a scale of 1 cm to 10 m.
Construct a line AE such that it bisects angle BAD and E is a point on the line BC
Construct the locus of points that are 40m from E
C
B
A
D
Shade the area of the field that is more than 40 m from E and closer to AB than AD