EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Enlargement
Transformations and constructions · Lesson 3 of 8
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
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
Learning Objectives
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).
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.
Enlarging from a Centre
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Lines through matching points all meet at the centre of enlargement. |
A triangle and its enlargement with scale factor 2 from the origin, with rays from the centre through matching vertices.
How to Enlarge a Shape
Work from the centre to each vertex.
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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. |
Enlarging from the Origin
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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).
Enlarging from Another Centre
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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).
A Fractional Scale Factor
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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.
Scale Factor Sizes
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SCALE FACTOR GREATER THAN 1 |
SCALE FACTOR BETWEEN 0 AND 1 |
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▸ 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. |
▸ The image is smaller than the object. ▸ Still called an enlargement. ▸ Example: scale factor ½ halves every length. |
HIGHER TIER
Negative Scale Factors and Similar Shapes
Enlarge through the centre, and how area and volume scale.
A Negative Scale Factor HIGHER
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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)
Lengths, Areas and Volumes HIGHER
For similar shapes with length scale factor k.
|
Measurement |
Scale factor |
|---|---|
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Lengths |
k |
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Areas |
k² |
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Volumes |
k³ |
Area and Volume Scale Factors HIGHER
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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.
Key Terms
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Enlargement A transformation that changes the size of a shape by a scale factor. |
Scale factor The number every length is multiplied by. |
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Centre of enlargement The fixed point the enlargement is measured from. |
Similar The same shape but a different size. |
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Ray A straight line drawn from the centre through a vertex. |
Congruent The same shape and size. |
Your Task: Describe the Enlargement
12 minutes
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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. |
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.
Note: Ask how to find the centre if it is not given (rays through matching vertices).
Can I...?
☐ 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).
Summary
✓ 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.
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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. |
Exam Practice: Enlargement
Answer all questions. Show your working. · 25 minutes
▸ Question 1 · 2 marks · Non-calculator. Triangle A has vertices (1, 1), (3, 1) and (1, 2). It is enlarged by scale factor 2 with centre (0, 0). Write down the coordinates of the…
▸ Question 2 · 2 marks · Non-calculator. A rectangle measures 3 cm by 5 cm. It is enlarged by scale factor 4. Write down the dimensions of the enlarged rectangle.
▸ Question 3 · 3 marks · Non-calculator. A triangle has vertices (2, 2), (4, 2) and (2, 3). It is enlarged by scale factor 2 with centre (1, 1). Write down the coordinates of the…
▸ Question 4 · 3 marks · Non-calculator. Triangle Q is an enlargement of triangle P. Describe fully the single transformation that maps P onto Q.
▸ Question 5 · 3 marks · Non-calculator. Two similar vases have heights 6 cm and 9 cm. The surface area of the smaller vase is 20 cm². Work out the surface area of the larger vase.
▸ Question 6 · 3 marks · Calculator. A model of a ship is made to a scale of 1 : 5. The volume of the model is 250 cm³. Work out the volume of the real ship in cm³.
Question 1 · 2 marks · Non-calculator
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“Triangle A has vertices (1, 1), (3, 1) and (1, 2). It is enlarged by scale factor 2 with centre (0, 0). Write down the coordinates of the vertices of the image.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 1 · mark scheme
2 marks available. Award a mark for each point made.
▸ Two vertices correct. B1
▸ All three correct. B1
▸ Model answer. (2, 2), (6, 2) and (2, 4).
Question 2 · 2 marks · Non-calculator
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“A rectangle measures 3 cm by 5 cm. It is enlarged by scale factor 4. Write down the dimensions of the enlarged rectangle.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 2 · mark scheme
2 marks available. Award a mark for each point made.
▸ One length correct. M1
▸ Both correct. A1
▸ Model answer. 12 cm by 20 cm.
Question 3 · 3 marks · Non-calculator
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“A triangle has vertices (2, 2), (4, 2) and (2, 3). It is enlarged by scale factor 2 with centre (1, 1). Write down the coordinates of the vertices of the image.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 3 · mark scheme
3 marks available. Award a mark for each point made.
▸ A correct method: vector from the centre doubled. M1
▸ Two vertices correct. A1
▸ All three correct. A1
▸ Model answer. (3, 3), (7, 3) and (3, 5).
Question 4 · 3 marks · Non-calculator
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Triangle Q is an enlargement of triangle P. Describe fully the single transformation that maps P onto Q. (3 marks) |
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Question 4 · mark scheme
3 marks available. Award a mark for each point made.
▸ Enlargement. B1
▸ Scale factor 2. B1
▸ Centre (0, 0). B1
▸ Model answer. An enlargement with scale factor 2 and centre (0, 0).
Question 5 · 3 marks · Non-calculator
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“Two similar vases have heights 6 cm and 9 cm. The surface area of the smaller vase is 20 cm². Work out the surface area of the larger vase.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 5 · mark scheme
3 marks available. Award a mark for each point made.
▸ Length scale factor 1.5. M1
▸ 1.5². M1
▸ 45. A1
▸ Model answer. The length scale factor is 9/6 = 1.5. The area scale factor is 1.5² = 2.25. The area is 20 × 2.25 = 45 cm².
Question 6 · 3 marks · Calculator
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“A model of a ship is made to a scale of 1 : 5. The volume of the model is 250 cm³. Work out the volume of the real ship in cm³.” |
HOW TO ANSWER IT Command word: Calculator. Worth 3 marks, so plan before writing.
Question 6 · mark scheme
3 marks available. Award a mark for each point made.
▸ Length scale factor 5. M1
▸ 5³ = 125. M1
▸ 31 250. A1
▸ Model answer. The length scale factor is 5, so the volume scale factor is 5³ = 125. The real volume is 250 × 125 = 31 250 cm³.