Loci Objectives: Grade DUnderstand the idea of a locus Grade CConstruct accurately loci, such as those of points equidistant from two fixed points Solve.

LociLoci

Objectives:

Grade D Understand the idea of a locus

Grade C Construct accurately loci, such as those of pointsequidistant from two fixed points

Solve loci problems, such as identifying pointsless than 3cm from a point P.

Prior knowledge:

Construct the perpendicular bisector of a line

Construct the bisector of an angle

LociLoci

A Locus is the path followed by a moving pointLoci is the plural of locus

1. The locus of points from a point P ....

xP

5 cm

... is a circle radius 5 cm, centre P

Loci can come in various forms:

LociLoci

2. The locus of points the same distance from line AB

A B

The locus of points 2 cm from line AB

2 cm

LociLoci

3. The locus of points equidistant from two fixed pointsA and B equidistant means:

‘same distance from’

A B

locus of points equidistant from A and B(perpendicular bisector of AB)

LociLoci

4. The locus of points equidistant from two fixed lines AB and AC

A

C

B

locus of points equidistant from AB and AC

(angle bisector of angle BAC)

LociLociImportant Note

For a rectangular shape a locus inside is different from a locus outside

Inside

Outside

Sharp corners

Rounded corners

LociLociWorked Example 1

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 borderScale: 1 large square = 1 metre

LociLociWorked 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 fromC and D. Mark the position of the treasure on the map.

A

B

D

C

locus of points equidistant from A and B

locus of pointsequidistant

from C and D

X

LociLociQuestion 1

8 cm

6 cm

86

LociLociQuestion 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

BA

LociLociQuestion 3A 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

LociLociQuestion 4The 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

AD

CB

Construct the locus of points that are 40m from E

Shade the area of the field that is more than 40 m from E and closer to AB than AD

E

Construct the angle bisector

Construct the locus around E4cm (40m)

LociLociWorked Example 1

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

LociLociWorked 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 fromC and D. Mark the position of the treasure on the map.

A

B

D

C

LociLociQuestion 1

86

LociLociQuestion 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

LociLociQuestion 3A 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

LociLociQuestion 4The 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

AD

CB

Construct the locus of points that are 40m from E

Shade the area of the field that is more than 40 m from E and closer to AB than AD