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10.4 Enlargement

GCSE Maths Foundation · Edexcel · Chapter 10: Transformations

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: scale factors, enlarging from a centre, and fractional scale factors. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Understand that an enlargement multiplies every length by the scale factor
  • Enlarge a shape by a whole-number scale factor, with or without a centre
  • Enlarge a shape by a fractional scale factor such as 1/2
  • Work out lengths and perimeters of enlarged shapes

Key words

  • Enlargement: A transformation that changes the size of a shape but not its shape. Angles stay the same.
  • Scale factor: The number every length is multiplied by.
  • Centre of enlargement: The fixed point the enlargement is measured from.
  • Similar: Two shapes are similar if one is an enlargement of the other.

Scale factors

In an enlargement every length is multiplied by the scale factor. The angles do not change, so the image is the same shape: it is similar to the object.

Worked example: lengths and perimeter

A rectangle 3 cm by 2 cm is enlarged by scale factor 3.
New lengths: 3 × 3 = 9 cm and 2 × 3 = 6 cm.
Perimeter before: 10 cm. Perimeter after: 30 cm, which is also 3 times as long.

Answer: 9 cm by 6 cm, perimeter 30 cm

Key fact: Scale factor = new length ÷ old length. A scale factor between 0 and 1 makes the shape smaller, but it is still called an enlargement.

Enlarging from a centre

With a centre of enlargement, the image is also placed in a fixed position. For each vertex, count the steps across and up from the centre, multiply them by the scale factor, and count that far from the centre.

Worked example: scale factor 3, centre (1, 1)

Vertex (4, 2) is 3 across and 1 up from C. Multiply by 3: 9 across and 3 up. Image: (1 + 9, 1 + 3) = (10, 4).
Vertex (2, 2) is 1 across, 1 up → 3 across, 3 up: (4, 4). Vertex (2, 3) is 1 across, 2 up → (4, 7).

Answer: T′ has vertices (4, 4), (10, 4) and (4, 7)

Check: Check: the lines from the centre through each object vertex pass through the matching image vertex.

Fractional scale factors

A scale factor of 1/2 halves every length, and the image is halfway between the centre and the object.

Worked example: scale factor 1/2, centre (1, 1)

Vertex (5, 3) is 4 across and 2 up from C. Half of that is 2 across and 1 up: (3, 2).
Vertex (7, 5) is 6 across, 4 up → 3 across, 2 up: (4, 3).

Answer: S′ has vertices (3, 2), (4, 2), (4, 3) and (3, 3)

Worked example: centre (0, 0)

With centre (0, 0) just multiply the coordinates by the scale factor.
(4, 6) enlarged by scale factor 1/2 → (2, 3). (2, −1) enlarged by scale factor 3 → (6, −3).

Answer: (2, 3) and (6, −3)

Common mistakes

  • Adding the scale factor instead of multiplying, e.g. a 3 cm side with scale factor 2 becomes 5 cm. It should be 6 cm.
  • Counting from the origin instead of from the given centre of enlargement.
  • Multiplying only some of the lengths. Every length is multiplied by the scale factor.
  • Thinking angles are multiplied too. Angles never change in an enlargement.

Revision questions

Try these, then check your answers below each one.

Question 1. A rectangle measures 5 cm by 2 cm. It is enlarged by scale factor 4. Work out the length of the longer side of the image.

Answer: 20 cm

Working: 5 × 4 = 20 cm.

Question 2. The point (3, −2) is enlarged by scale factor 2, centre (0, 0). Write down the coordinates of its image.

Answer: (6,-4)

Working: (3 × 2, -2 × 2) = (6, -4).

Question 3. A triangle is enlarged by scale factor 3. What happens to its angles? (Choose from: They stay the same, They are multiplied by 3, They are divided by 3, They add up to 540°.)

Answer: They stay the same

Working: Only lengths change. The angles stay the same, so the image is similar to the object.

Related lessons

Continue in the full course

This lesson is section 10.4 (EGCMF1004) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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