Constructions 2 - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Constructions 2

Transformations and constructions

Lesson 7 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Transformations and constructions

Lesson 7 of 8

Answer each one, then check.

1

What does perpendicular mean?

2

What is half of 60°?

3

How many degrees in each angle of an equilateral triangle?

4

What is an angle bisector?

5

What are the tools for a construction?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Transformations and constructions

Lesson 7 of 8

1

What does perpendicular mean?

At 90°

2

What is half of 60°?

30°

3

How many degrees in each angle of an equilateral triangle?

60°

4

What is an angle bisector?

A line that cuts an angle in half

5

What are the tools for a construction?

Ruler, compasses (and a sharp pencil)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Transformations and constructions

Lesson 7 of 8

1

Construct the bisector of an angle.

2

Construct a perpendicular from a point to a line.

3

Construct a perpendicular at a point on a line.

4

Construct angles of 60°, 30° and 90°.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Bisecting an Angle

Transformations and constructions

Lesson 7 of 8

The bisector splits the angle into two equal halves.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Bisecting an Angle

Transformations and constructions

Lesson 7 of 8

Three arcs and one line.

1

Arc from the vertex

Centre B, cutting both arms at P and Q

2

Arcs from P and Q

Same radius from each; they cross at R

3

Draw the line BR

This is the angle bisector

4

Check

Measure the two halves; they should be equal

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Angle Bisector Check

Transformations and constructions

Lesson 7 of 8

Angle ABC is 70°. After bisecting it, what is the size of each half?

1

The bisector cuts the angle in half

70 ÷ 2

2

Work out

35

ANSWER

Each half is 35°.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Perpendiculars

Transformations and constructions

Lesson 7 of 8

Both use the perpendicular bisector method on a pair of points.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Perpendicular from a Point to a Line

Transformations and constructions

Lesson 7 of 8

Point P is above the line.

1

Arc centred on P

It crosses the line at two points, X and Y

2

Arcs from X and Y

Same radius, crossing on the other side of the line at Z

3

Join P to Z

This line is perpendicular to the line

4

Label the right angle

Mark it with a small square

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Perpendicular at a Point on a Line

Transformations and constructions

Lesson 7 of 8

Point Q is on the line.

1

Arc centred on Q

It crosses the line either side at U and V

2

Larger arcs from U and V

Same radius, crossing above the line at W

3

Join Q to W

This line is perpendicular to the line at Q

4

Check

The angle is 90°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Constructing 60° and 30°

Transformations and constructions

Lesson 7 of 8

Compasses can make an equilateral triangle, which gives 60°.

60°

Draw an arc from A across the line at B, then an arc of the same radius from B; join A to where they cross.

30°

Bisect the 60° angle.

90°

Use the perpendicular construction.

45°

Bisect a 90° angle.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Transformations and constructions

Lesson 7 of 8

Angle bisector

A line that cuts an angle exactly in half.

Perpendicular

At 90° to a line.

Arc

Part of a circle's circumference.

Equidistant

The same distance from two things.

Construction lines

The arcs and lines you leave to show your method.

Vertex

The point where the arms of an angle meet.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 7 of 8

YOUR TASK

Build a Kite

12 minutes

Construct an angle of 60° at A. Bisect it to make 30°. Then use the perpendicular construction to make a right angle at a point on one arm. Sketch a kite using your construction lines.

1

Make 60° with compasses.

2

Bisect it.

3

Add a 90° angle.

WHAT A GOOD ANSWER SHOWS

A 60° angle from an equilateral construction, 30° from the bisector, and a 90° angle by the perpendicular at a point. A kite can be made from two triangles sharing the bisector.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Transformations and constructions

Lesson 7 of 8

Bisect an angle.

Drop a perpendicular from a point.

Draw a perpendicular at a point on a line.

Construct 60°.

Construct 30°.

Construct 90°.

Explain why the constructions work.

Show all construction lines.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Angle bisector: arc, two equal arcs, join to the vertex.

Perpendicular from a point: arc, two equal arcs, join.

60° from an equilateral triangle; halve it for 30°.

Keep all construction arcs.

EXAM FOCUS

Using ruler and compasses only, construct the bisector of angle ABC. You must show all your construction lines. (2 marks)

Start with an arc centred on the vertex, then use equal arcs from the two points where it crosses the arms.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Constructions 2

Transformations and constructions

Lesson 7 of 8

Answer all questions. Use a sharp pencil, a ruler and compasses. Show all construction lines. · 30 minutes

Question 1 · 2 marks · Construct

The diagram shows an angle ABC. Using ruler and compasses only, construct the bisector of angle ABC. Show all your construction lines.

Question 2 · 3 marks · Construct

Point P is 4 cm above a horizontal line. Using ruler and compasses only, construct the perpendicular from P to the line. Show all your…

Question 3 · 3 marks · Construct

Using ruler and compasses only, construct an angle of 60° at the end A of a line AB.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Constructions 2 (cont.)

Transformations and constructions

Lesson 7 of 8

Question 4 · 3 marks · Construct

Using ruler and compasses only, construct an angle of 30° at the end A of a line AB.

Question 5 · 3 marks · Construct

Q is a point on a straight line. Using ruler and compasses only, construct the perpendicular to the line at Q.

Question 6 · 2 marks · Explain

Explain why the line through the crossing arcs is the bisector of the angle in the angle bisector construction.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 2 marks · Construct

Transformations and constructions

Lesson 7 of 8

The diagram shows an angle ABC. Using ruler and compasses only, construct the bisector of angle ABC. Show all your construction lines. (2 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Transformations and constructions

Lesson 7 of 8

2 marks available. Award a mark for each point made.

Arcs from B and from the two crossing points

M1

A correct bisector with arcs shown

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 3 marks · Construct

Transformations and constructions

Lesson 7 of 8

“

Point P is 4 cm above a horizontal line. Using ruler and compasses only, construct the perpendicular from P to the line. Show all your construction lines.

HOW TO ANSWER IT

Command word: Construct. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Transformations and constructions

Lesson 7 of 8

3 marks available. Award a mark for each point made.

Arc from P crossing the line twice

M1

Equal arcs from the two points

M1

A correct perpendicular

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 3 marks · Construct

Transformations and constructions

Lesson 7 of 8

“

Using ruler and compasses only, construct an angle of 60° at the end A of a line AB.

HOW TO ANSWER IT

Command word: Construct. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Transformations and constructions

Lesson 7 of 8

3 marks available. Award a mark for each point made.

Arc centred on A

M1

Second arc with the same radius

M1

Line completed and angle correct

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 3 marks · Construct

Transformations and constructions

Lesson 7 of 8

“

Using ruler and compasses only, construct an angle of 30° at the end A of a line AB.

HOW TO ANSWER IT

Command word: Construct. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Transformations and constructions

Lesson 7 of 8

3 marks available. Award a mark for each point made.

60° constructed

M1

Bisector constructed

M1

Angle of 30°

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 3 marks · Construct

Transformations and constructions

Lesson 7 of 8

“

Q is a point on a straight line. Using ruler and compasses only, construct the perpendicular to the line at Q.

HOW TO ANSWER IT

Command word: Construct. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Transformations and constructions

Lesson 7 of 8

3 marks available. Award a mark for each point made.

Arc centred on Q

M1

Equal arcs from both points

M1

Correct perpendicular

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 2 marks · Explain

Transformations and constructions

Lesson 7 of 8

“

Explain why the line through the crossing arcs is the bisector of the angle in the angle bisector construction.

HOW TO ANSWER IT

Command word: Explain. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Transformations and constructions

Lesson 7 of 8

2 marks available. Award a mark for each point made.

Equal distances used

M1

Congruent triangles or kite symmetry

C1

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