Enlargement.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Enlargement

Transformations and constructions

Lesson 3 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Transformations and constructions

Lesson 3 of 8

Answer each one, then check.

1

What is 3 × 4?

2

Work out ½ × 8.

3

What is the area of a rectangle 3 by 5?

4

What is the scale factor from 4 cm to 12 cm?

5

What does congruent mean?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Transformations and constructions

Lesson 3 of 8

1

What is 3 × 4?

12

2

Work out ½ × 8.

4

3

What is the area of a rectangle 3 by 5?

15

4

What is the scale factor from 4 cm to 12 cm?

3

5

What does congruent mean?

The same shape and size

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Transformations and constructions

Lesson 3 of 8

1

Enlarge a shape by a positive scale factor from a centre.

2

Use fractional scale factors.

3

Describe an enlargement fully.

4

Use negative scale factors and similar-shape area and volume ratios (Higher).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 3 of 8

THE KEY IDEA

An enlargement changes the size of a shape but not its shape: every length is multiplied by the scale factor.

The centre of enlargement stays in the same place.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Enlarging from a Centre

Transformations and constructions

Lesson 3 of 8

Lines through matching points all meet at the centre of enlargement.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

How to Enlarge a Shape

Transformations and constructions

Lesson 3 of 8

Work from the centre to each vertex.

1

Pick the centre

It is given, or found where lines through matching points meet.

2

Measure to a vertex

Count squares across and up from the centre.

3

Multiply by the scale factor

Multiply both distances.

4

Plot the new vertex

Start again from the centre.

5

Join up

Join the new vertices to make the image.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Enlarging from the Origin

Transformations and constructions

Lesson 3 of 8

Triangle A has vertices (1, 1), (3, 1) and (1, 2). Enlarge it by scale factor 2 with centre (0, 0).

1

Multiply each coordinate by 2

(1, 1) → (2, 2)

2

The other vertices

(3, 1) → (6, 2) and (1, 2) → (2, 4)

ANSWER

The image has vertices (2, 2), (6, 2) and (2, 4).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Enlarging from Another Centre

Transformations and constructions

Lesson 3 of 8

A triangle has vertices (2, 2), (4, 2) and (2, 3). Enlarge it by scale factor 2 about the centre (1, 1).

1

Vector from the centre to the first vertex

(2, 2) − (1, 1) = (1, 1)

2

Multiply by 2

(2, 2)

3

Add back to the centre

(1 + 2, 1 + 2) = (3, 3)

4

The other vertices

(4, 2): vector (3, 1), doubled (6, 2), so (7, 3); (2, 3): vector (1, 2), doubled (2, 4), so (3, 5)

ANSWER

The image has vertices (3, 3), (7, 3) and (3, 5).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Fractional Scale Factor

Transformations and constructions

Lesson 3 of 8

Enlarge the triangle with vertices (2, 4), (6, 4) and (2, 8) by scale factor ½ with centre (0, 0).

1

Multiply each coordinate by ½

(2, 4) → (1, 2)

2

The other vertices

(6, 4) → (3, 2) and (2, 8) → (1, 4)

ANSWER

The image has vertices (1, 2), (3, 2) and (1, 4); it is smaller than the original.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Scale Factor Sizes

Transformations and constructions

Lesson 3 of 8

SCALE FACTOR GREATER THAN 1

• The image is bigger than the object.

• Lengths are multiplied by the scale factor.

• Example: scale factor 3 makes every side three times as long.

SCALE FACTOR BETWEEN 0 AND 1

• The image is smaller than the object.

• Still called an enlargement.

• Example: scale factor ½ halves every length.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

HIGHER TIER

Negative Scale Factors and Similar Shapes

Enlarge through the centre, and how area and volume scale.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

A Negative Scale Factor

Transformations and constructions

Lesson 3 of 8

Enlarge the point (1, 2) by scale factor −2 about the origin.

1

Multiply the coordinates by −2

(1, 2) → (−2, −4)

2

The image is on the opposite side of the centre

And twice as far away

ANSWER

(−2, −4)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

Lengths, Areas and Volumes

Transformations and constructions

Lesson 3 of 8

For similar shapes with length scale factor k.

Measurement

Scale factor

Lengths

k

Areas

k²

Volumes

k³

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

Area and Volume Scale Factors

Transformations and constructions

Lesson 3 of 8

Two similar bottles have heights 6 cm and 9 cm. The label of the smaller bottle has area 20 cm². Find the label area on the larger bottle. The smaller bottle holds 250 ml; find the capacity of the larger.

1

Length scale factor

9 ÷ 6 = 1.5

2

Area scale factor

1.5² = 2.25, so 20 × 2.25 = 45

3

Volume scale factor

1.5³ = 3.375, so 250 × 3.375 = 843.75

ANSWER

Label area 45 cm² and capacity 843.75 ml.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Transformations and constructions

Lesson 3 of 8

Enlargement

A transformation that changes the size of a shape by a scale factor.

Scale factor

The number every length is multiplied by.

Centre of enlargement

The fixed point the enlargement is measured from.

Similar

The same shape but a different size.

Ray

A straight line drawn from the centre through a vertex.

Congruent

The same shape and size.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 3 of 8

YOUR TASK

Describe the Enlargement

12 minutes

Triangle P has vertices (2, 1), (4, 1) and (2, 4). Triangle Q has vertices (4, 2), (8, 2) and (4, 8). Describe fully the single transformation that maps P onto Q. Then find the ratio of their areas.

1

Compare corresponding side lengths.

2

Draw rays to find the centre.

3

Compare areas.

WHAT A GOOD ANSWER SHOWS

Enlargement, scale factor 2, centre (0, 0). The area of P is 3 and the area of Q is 12, a ratio of 1:4, which is 2².

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Transformations and constructions

Lesson 3 of 8

Enlarge from the origin.

Enlarge from another centre.

Use a fractional scale factor.

Describe an enlargement fully.

Find the centre using rays.

Use a negative scale factor (Higher).

Use k² for areas (Higher).

Use k³ for volumes (Higher).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Every length is multiplied by the scale factor.

Scale factors below 1 make the image smaller.

To describe: enlargement, scale factor, centre.

Higher: areas scale by k² and volumes by k³.

EXAM FOCUS

Describe fully the single transformation that maps triangle P onto triangle Q, where P has vertices (2, 1), (4, 1), (2, 4) and Q has vertices (4, 2), (8, 2), (4, 8). (3 marks)

For an enlargement give three things: the word enlargement, the scale factor, and the centre.

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