Loci Objectives: Grade DUnderstand the idea of a locus Grade CConstruct accurately loci, such as...
-
Upload
lilian-shaw -
Category
Documents
-
view
231 -
download
0
Transcript of Loci Objectives: Grade DUnderstand the idea of a locus Grade CConstruct accurately loci, such as...
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