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Maths · Transformations and Similarity

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Enlargements

Enlarging shapes by whole-number, fractional and negative scale factors, and describing an enlargement fully.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Enlarge a shape by a positive whole-number or fractional scale factor from a given centre.
  2. 2Describe an enlargement fully by its scale factor and centre.
  3. 3Find the scale factor and centre of an enlargement from a diagram.
  4. 4Enlarge a shape by a negative scale factor (Higher tier), and say what happens to lengths and angles.

Changing the size

An enlargement changes the size of a shape, but not its angles, so the image is similar to the object but not congruent unless the scale factor is 1. Every length is multiplied by the scale factor, and the centre of enlargement is the fixed point from which the distances are scaled. A scale factor bigger than 1 makes the shape larger, and a scale factor between 0 and 1 makes it smaller, which is still called an enlargement. Enlargements are tested on the non-calculator paper with simple numbers on a grid, and the marks come from the centre as well as the scale factor.

Scale factor and centre

Every length is multiplied by the scale factor, and every point moves along a line from the centre.

  • Lengths

    A side of 3 cm enlarged by a scale factor of 2 becomes 6 cm.

  • Angles

    They stay the same.

  • Distances from the centre

    The image of a point is on the line from the centre through the point, at a distance of scale factor times the original distance.

  • Fractional scale factors

    A scale factor of \(\dfrac{1}{2}\) halves every length, so the image is smaller and closer to the centre.

Enlarging a point

The centre of enlargement is \((1, 2)\) and the scale factor is 3. Find the image of the point \((3, 3)\).

Show the solutionHide the solution
  1. 1 Distance from the centre \((3, 3)\) is 2 right and 1 up from \((1, 2)\).
  2. 2 Multiply by 3 The image is \(3 \times 2 = 6\) right and \(3 \times 1 = 3\) up from the centre.
  3. 3 Count from the centre \((1 + 6, 2 + 3) = (7, 5)\).
  4. 4 Check The centre, \((3, 3)\) and \((7, 5)\) are on a line, because the steps \((2, 1)\) and \((6, 3)\) have the same direction.

Answer\((7, 5)\)

Describing an enlargement

Give the scale factor and the centre, and name the transformation.

  • Scale factor

    Divide a length on the image by the matching length on the object. A side of 6 cm from 2 cm gives a scale factor of 3.

  • Centre

    Draw a line through each corner and its image, and the lines meet at the centre.

  • A small image

    The scale factor is less than 1, such as \(\dfrac{1}{2}\), and the image is closer to the centre.

  • Full description

    "Enlargement, scale factor \(2\), centre \((1, 1)\)" gives all the marks.

Finding the scale factor and centre

A rectangle with corners \((2, 2)\), \((4, 2)\), \((4, 3)\) and \((2, 3)\) is enlarged to give a rectangle with corners \((3, 3)\), \((7, 3)\), \((7, 5)\) and \((3, 5)\). Describe the enlargement.

Show the solutionHide the solution
  1. 1 Scale factor The bottom of the object is 2 long and the bottom of the image is 4 long, so the scale factor is 2.
  2. 2 Centre Join \((2, 2)\) to \((3, 3)\) and \((4, 3)\) to \((7, 5)\), and extend the lines back.
  3. 3 They meet at The point \((1, 1)\).
  4. 4 Check \((1, 1)\) to \((2, 2)\) is 1 right and 1 up, and \((1, 1)\) to \((3, 3)\) is twice that.

AnswerEnlargement, scale factor 2, centre \((1, 1)\)

Negative scale factors (Higher tier)

A negative scale factor puts the image on the other side of the centre, and turns it upside down.

  • The image is on the opposite side

    A point 2 right and 1 up from the centre goes to 2 left and 1 down for a scale factor of \(-1\).

  • Size

    The lengths are multiplied by the size of the scale factor, so \(-2\) gives sides twice as long.

  • Same as a rotation

    An enlargement with scale factor \(-1\) is a rotation of \(180^\circ\) about the centre.

  • Describe

    Give the negative scale factor and the centre, such as "enlargement, scale factor \(-2\), centre \((0, 0)\)".

Test yourself

  1. 1

    What happens to the angles in an enlargement?

    Show answerHide answer

    They stay the same.

  2. 2

    What is the scale factor from 3 cm to 12 cm?

    Show answerHide answer

    4.

  3. 3

    What do you give to describe an enlargement?

    Show answerHide answer

    The scale factor and the centre.

  4. 4

    What does a scale factor between 0 and 1 do?

    Show answerHide answer

    It makes the shape smaller.

  5. 5

    Which way is the image for a negative scale factor?

    Show answerHide answer

    On the opposite side of the centre.

Exam technique: enlargements

Use the centre, and check with straight lines.

  • Count from the centre, not from the shape

    Distances are measured from the centre of enlargement.

  • Draw construction lines

    Rays through the corners help the examiner see your method.

  • Check two corners

    They should both be on lines from the centre.

  • Use the right words

    "Enlargement" with a scale factor and a centre, not "bigger" or "zoom".

Summary and exam focus

  • An enlargement multiplies every length by the scale factor, and keeps the angles.
  • The image of a point is on the line from the centre through the point, at scale factor times the distance.
  • A full description needs the scale factor and the centre.
  • A negative scale factor (Higher tier) puts the image on the other side of the centre.

Exam focus

The point \(A\) is \((2, 3)\). Enlarge the point \(A\) by scale factor 2 with centre \((0, 0)\), and write down the coordinates of the image. (2 marks) (2 marks)

Multiply the coordinates by the scale factor, since the centre is the origin: \((2 \times 2, 3 \times 2) = (4, 6)\). With a different centre, count from the centre instead.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Enlargement
A transformation that changes the size of a shape by a scale factor.
Scale factor
The number by which every length of a shape is multiplied.
Centre of enlargement
The fixed point from which an enlargement is measured.
Similar
The same shape but not necessarily the same size.
Congruent
Having exactly the same size and shape.
Corresponding sides
Sides in the same position on two similar shapes.
Image
The shape after a transformation.
Ray
A straight line drawn from the centre through a point and its image.
Negative scale factor
A scale factor that puts the image on the opposite side of the centre.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Write down 2 marks Easier

A rectangle is 3 cm by 5 cm. It is enlarged by a scale factor of 4. Write down the length and the width of the enlarged rectangle.

Mark scheme — 2 marks available

  • \(3 \times 4\) or \(5 \times 4\) — M1
  • 12 cm by 20 cm — A1

Model answer

Each length is multiplied by 4, so the enlarged rectangle is \(3 \times 4 = 12\) cm by \(5 \times 4 = 20\) cm.

2. Exam question Enlarge 3 marks Core

Enlarge triangle \(T\) by a scale factor of 3 with the centre of enlargement \(O\), the origin. (3 marks)

Triangle T on a grid with the centre of enlargement O at the origin.

Mark scheme — 3 marks available

  • Enlarges at least two vertices by a scale factor of 3 — M1
  • At least two vertices correct — A1
  • Triangle with vertices \((3, 3)\), \((9, 3)\) and \((3, 6)\) — A1

Model answer

Multiply each coordinate by 3: \((1, 1)\) goes to \((3, 3)\), \((3, 1)\) goes to \((9, 3)\) and \((1, 2)\) goes to \((3, 6)\).

3. Exam question Describe 3 marks Core

Describe fully the single transformation that maps triangle \(A\) onto triangle \(B\).

Triangle A and its image triangle B on a grid.

Mark scheme — 3 marks available

  • Enlargement — B1
  • Scale factor 2 — B1
  • Centre \((1, 0)\) — B1

Model answer

The bottom of \(A\) is 2 long and the bottom of \(B\) is 4 long, so the scale factor is 2. The lines through matching corners, such as \((2, 1)\) and \((3, 2)\), and \((4, 1)\) and \((7, 2)\), meet at \((1, 0)\). So it is an enlargement, scale factor 2, centre \((1, 0)\).

4. Exam question Work out 3 marks Core

A triangle has sides of 4 cm, 6 cm and 8 cm. It is enlarged by a scale factor of 1.5. (a) Work out the lengths of the sides of the enlarged triangle. (2 marks) (b) Work out the perimeter of the enlarged triangle. (1 mark)

Mark scheme — 3 marks available

  • (a) At least one length multiplied by 1.5 — M1
  • (a) 6 cm, 9 cm and 12 cm — A1
  • (b) 27 cm — B1 (follow through from (a))

Model answer

(a) \(4 \times 1.5 = 6\), \(6 \times 1.5 = 9\) and \(8 \times 1.5 = 12\). (b) \(6 + 9 + 12 = 27\) cm.

5. Exam question Work out 3 marks Core

A square has sides of 12 cm. It is enlarged by a scale factor of \(\dfrac{1}{3}\). (a) Write down the side length of the enlarged square. (1 mark) (b) Work out the area of the enlarged square. (2 marks)

Mark scheme — 3 marks available

  • (a) 4 cm — B1
  • (b) \(4 \times 4\) — M1
  • (b) 16 cm\(^2\) — A1 (follow through from (a))

Model answer

(a) \(12 \times \dfrac{1}{3} = 4\) cm. (b) \(4 \times 4 = 16\) cm\(^2\).

6. Exam question Work out 3 marks Stretch

The point \(A\) is \((3, 2)\). \(A\) is enlarged by a scale factor of \(-2\) with the centre \((1, 1)\). Work out the coordinates of the image of \(A\).

Mark scheme — 3 marks available

  • Position relative to the centre, \((2, 1)\) — M1
  • \((-4, -2)\) relative to the centre — M1
  • \((-3, -1)\) — A1

Model answer

\(A\) is 2 right and 1 up from the centre. With a scale factor of \(-2\), the image is 4 left and 2 down from the centre, at \((1 - 4, 1 - 2) = (-3, -1)\).

7. Multiple choice 1 mark Easier

A side of 3 cm is enlarged to 12 cm. What is the scale factor?

  1. A \(9\)
  2. B \(\dfrac{1}{4}\)
  3. C \(36\)
  4. D \(4\) Correct

Why: \(\dfrac{12}{3} = 4\).

8. Multiple choice 1 mark Easier

The point \((2, 3)\) is enlarged by scale factor 3 with centre \((0, 0)\). What is the image?

  1. A \((5, 6)\)
  2. B \((6, 3)\)
  3. C \((6, 9)\) Correct
  4. D \((9, 6)\)

Why: Multiply both coordinates by 3.

9. Multiple choice 1 mark Easier

What happens to the angles in an enlargement?

  1. A They are multiplied by the scale factor
  2. B They stay the same Correct
  3. C They are halved
  4. D They are added to the scale factor

Why: An enlargement keeps the angles, so the shapes are similar.

10. Multiple choice 1 mark Easier

What does an enlargement with scale factor \(\dfrac{1}{2}\) do to a shape?

  1. A It halves every length Correct
  2. B It doubles every length
  3. C It halves every angle
  4. D It leaves the shape unchanged

Why: A fractional scale factor less than 1 makes the shape smaller.

11. Multiple choice 1 mark Core

The centre of enlargement is \((1, 2)\) and the scale factor is 3. What is the image of \((3, 3)\)?

  1. A \((9, 9)\)
  2. B \((5, 4)\)
  3. C \((6, 5)\)
  4. D \((7, 5)\) Correct

Why: \((3, 3)\) is 2 right and 1 up from the centre, so the image is 6 right and 3 up: \((7, 5)\).

12. Multiple choice 1 mark Easier

What must you give to describe an enlargement fully?

  1. A The mirror line
  2. B The angle and the direction
  3. C The scale factor and the centre Correct
  4. D A column vector

Why: An enlargement is described by its scale factor and its centre.

13. Multiple choice 1 mark Core

A triangle with sides 4, 6 and 8 is enlarged to give a triangle with sides 10, 15 and 20. What is the scale factor?

  1. A \(2\)
  2. B \(2.5\) Correct
  3. C \(6\)
  4. D \(3\)

Why: \(\dfrac{10}{4} = 2.5\).

14. Multiple choice 1 mark Stretch

Which transformation is the same as an enlargement with scale factor \(-1\)?

  1. A A rotation of \(180^\circ\) about the centre Correct
  2. B A reflection in a line through the centre
  3. C A translation
  4. D A rotation of \(90^\circ\)

Why: Every point goes to the opposite side of the centre at the same distance.

15. Multiple choice 1 mark Stretch

The point \((1, 3)\) is enlarged by scale factor \(-2\) with centre \((0, 0)\). What is the image?

  1. A \((2, 6)\)
  2. B \((-1, -3)\)
  3. C \((-2, 6)\)
  4. D \((-2, -6)\) Correct

Why: Multiply both coordinates by \(-2\).