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Mastering Loci: Understanding and Solving Loci Problems Effectively

Learn the concept of loci, construct accurate loci, and solve problems like identifying equidistant points. Explore loci in various forms and tackle loci questions with confidence. Worked examples included for better understanding.

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Mastering Loci: Understanding and Solving Loci Problems Effectively

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  1. 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

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

  3. 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

  4. Loci 3. The locus of points equidistant from two fixed points A and B equidistant means: ‘same distance from’ A B locus of points equidistant from A and B (perpendicular bisector of AB)

  5. 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)

  6. Loci Important Note For a rectangular shape a locus inside is different from a locus outside Inside Sharp corners Outside Rounded corners

  7. Loci Worked 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 border Scale: 1 large square = 1 metre

  8. 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

  9. Loci Question 1 8 6 8 cm 6 cm

  10. 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 B A

  11. Loci Question 3 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

  12. 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 C B 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

  13. Loci Worked 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

  14. 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

  15. Loci Question 1 8 6

  16. 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

  17. Loci Question 3 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

  18. 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

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