Loci Objectives Grade D Understand the idea of

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 3 cm 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 5 m 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 3 m. Sprinkler A is 3 m from the top edge and 3 m from the left edge. Sprinkler B is 3 m from the top edge and 3 m from the right edge. Sprinkler C is 3 m from the bottom edge and 6 m 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 40 m from E Construct the locus around E 4 cm (40 m) 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 5 m 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 3 m. Sprinkler A is 3 m from the top edge and 3 m from the left edge. Sprinkler B is 3 m from the top edge and 3 m from the right edge. Sprinkler C is 3 m from the bottom edge and 6 m 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 40 m 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