Loci.pptx — OpenRevise

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Loci

Transformations and constructions

Lesson 8 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Transformations and constructions

Lesson 8 of 8

Answer each one, then check.

1

What is the distance round a circle called?

2

What is a perpendicular bisector?

3

What does an angle bisector do?

4

What is the radius of a circle with diameter 8 cm?

5

What does equidistant mean?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Transformations and constructions

Lesson 8 of 8

1

What is the distance round a circle called?

The circumference

2

What is a perpendicular bisector?

A line at 90° through the midpoint of a segment

3

What does an angle bisector do?

Cuts an angle in half

4

What is the radius of a circle with diameter 8 cm?

4 cm

5

What does equidistant mean?

The same distance from

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Transformations and constructions

Lesson 8 of 8

1

Explain what a locus is.

2

Draw the four standard loci.

3

Combine loci to find a region.

4

Solve problems in context, using a scale.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 8 of 8

LOCUS

A locus is the set of all points that follow a rule.

The plural is loci.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Standard Loci

Transformations and constructions

Lesson 8 of 8

Learn these four and you can solve most locus questions.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Four Standard Loci

Transformations and constructions

Lesson 8 of 8

A fixed distance from a point

A circle, centred on the point.

Equidistant from two points

The perpendicular bisector of the line joining the points.

Equidistant from two lines

The angle bisector of the angle between them.

A fixed distance from a line

A racetrack: two parallel lines with semicircular ends.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Region Bounded by Two Loci

Transformations and constructions

Lesson 8 of 8

ABCD is a rectangle with A at (0, 0), B at (8, 0), C at (8, 5) and D at (0, 5), with 1 cm representing 1 m. A tree is nearer to AB than to AD and less than 4 m from C. Describe the region where the tree can be.

1

Equidistant from AB and AD

The bisector of angle DAB, the line from A at 45°

2

Nearer to AB than AD

The side of the bisector next to AB (below the line y = x)

3

Less than 4 m from C

Inside a circle of radius 4 cm centred on C

4

Combine

The part of the rectangle that is both below the bisector and inside the circle

ANSWER

The region inside the rectangle, below the bisector from A and inside the circle of radius 4 cm centred on C.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Tackling a Locus Question

Transformations and constructions

Lesson 8 of 8

Split the rule into simple loci.

1

Read the rule

Underline each condition.

2

Draw the standard locus for each

Circle, bisector, perpendicular bisector, or parallel lines.

3

Decide which side

Use a test point.

4

Shade the region

Or mark the route required.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equidistant from Two Points

Transformations and constructions

Lesson 8 of 8

P and Q are 6 cm apart. Draw the locus of points equidistant from P and Q, and within 4 cm of P.

1

The locus of points equidistant from P and Q

The perpendicular bisector of PQ

2

Within 4 cm of P

Inside a circle of radius 4 cm centred on P

3

The part of the bisector inside the circle

A line segment where the bisector crosses the circle

ANSWER

The segment of the perpendicular bisector of PQ that lies inside the circle of radius 4 cm centred on P.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Fixed Distance from a Line

Transformations and constructions

Lesson 8 of 8

A path is 6 m long. A dog is allowed within 2 m of the path. Describe the boundary of the area.

1

The set of points 2 m from a line

Two parallel lines, 2 m either side

2

Ends of the path

Two semicircles of radius 2 m

3

Together

A racetrack shape around the path

ANSWER

A racetrack: two parallel lines 2 m either side of the path with a semicircle of radius 2 m at each end.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Transformations and constructions

Lesson 8 of 8

Locus

A set of points that all obey a rule.

Loci

More than one locus.

Equidistant

The same distance from two points or lines.

Region

An area of the plane described by one or more rules.

Perpendicular bisector

The locus of points equidistant from two points.

Angle bisector

The locus of points equidistant from two lines.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 8 of 8

YOUR TASK

Where Should the Mast Go?

15 minutes

Three villages A, B and C form a triangle. A phone mast must be equidistant from A and B, and no more than 5 km from C. Draw the triangle (AB = 8 cm, AC = 6 cm, BC = 7 cm, where 1 cm = 1 km), construct the perpendicular bisector of AB, draw the circle of radius 5 cm about C, and shade the possible positions.

1

Construct the triangle.

2

Draw the two loci.

3

Identify the overlap.

WHAT A GOOD ANSWER SHOWS

The possible positions lie along the segment of the perpendicular bisector of AB that lies within the circle. Students should show construction arcs for the bisector and mark the two points where it meets the circle.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Transformations and constructions

Lesson 8 of 8

Explain what a locus is.

Draw a circle locus.

Draw a perpendicular bisector locus.

Draw an angle bisector locus.

Draw a racetrack locus.

Combine two loci.

Use a scale in a locus problem.

Shade the correct region.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Circle: a fixed distance from a point.

Perpendicular bisector: equidistant from two points.

Angle bisector: equidistant from two lines.

Racetrack: a fixed distance from a line segment.

EXAM FOCUS

ABCD is a rectangle with AB = 8 cm and BC = 5 cm. A point is nearer to AB than to AD, and is less than 4 cm from C. Shade the region that contains all such points. (4 marks)

Draw each locus, then use one test point in the rectangle to decide which side of each locus to shade.

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