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Enlargements (Cambridge O Level Maths)

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Enlargements

What is an enlargement?

  • An enlargement is a transformation that changes the size of the shape
    • The scale factor tells you how many times bigger each edge of the enlarged image will be compared to the corresponding edge on the original object
    • If the scale factor is greater than 1, the enlarged image will be bigger than the original object
    • If the scale factor is less than 1, the enlarged image will be smaller than the original object
      • See Fractional Enlargements
  • The position of a shape will also change with enlargement
    • This is determined by the centre of enlargement (CoE)
    • The centre of enlargement is a coordinate point on the grid
  • The orientation of the shape will be the same for a positive enlargement
    • We do not need to consider negative enlargements for this course

How do I enlarge a shape?

  • You need to be able to perform an enlargement (on a coordinate grid)

  • STEP 1

    Starting from the centre of enlargement, count the horizontal and vertical distances to any one vertex on the original shape

  • STEP 2

    Multiply both the horizontal and vertical distances by the given scale factor

  • STEP 3
    Starting again from the centre of enlargement, and counting in the same directions
    • measure these new (multiplied) horizontal and vertical distances
    • mark the position on the grid of the corresponding vertex on the enlarged image
  • STEP 4
    Repeat Steps 1 to 3 for the remaining vertices
    • You may not need to do this for every vertex, if you are confident you can draw the enlarged shape once a couple of vertices are in place
    • We'd recommend you do at least two vertices though to be sure 
  • STEP 5
    Connect the vertices on the enlarged image and label it

How do I describe an enlargement?

  • You need to be able to identify and describe an enlargement when presented with one
  • You must fully describe a transformation to get full marks
  • For an enlargement, you must:
    • State that the transformation is an enlargement
    • State the scale factor
    • Give the coordinates of the centre of enlargement

Examiner Tip

  • Make sure that you always start from the centre of enlargement
    • this is both for measuring distances to the original object and then measuring for the enlarged image
    • a common mistake is to measure the distance between a pair of corresponding vertices on the original object and enlarged image
  • You can check your work by drawing straight lines through the centre of enlargement and a pair of corresponding vertices on the original object and the enlarged image
    • All such lines should intersect at the centre of enlargement

Worked example

a)
On the grid below enlarge shape C using scale factor 2 and centre of enlargement (2, 1).
Label your enlarged shape C'.

Enlargement-Q1, IGCSE & GCSE Maths revision notes

Start by marking on the centre of enlargement (CoE).
Count the number of squares in both a horizontal and vertical direction to go from the CoE to one of the vertices on the original object, this is 2 to the right and 3 up in this example.
As the scale factor is 2, multiply these distances by 2, so they become 4 to the right and 6 up.
Count these new distances from the CoE to the corresponding point on the enlarged image and mark it on.
Draw a line through the CoE and the pair of corresponding points, they should line up in a straight line. 

Enlargement-Q1-working, IGCSE & GCSE Maths revision notes

Repeat this process for each of the vertices on the original object (or at least 2).
Join adjacent vertices on the enlarged image as you go.
Label the enlarged image C'.

Enlargement-Q1-Final-Answer, IGCSE & GCSE Maths revision notes

b)
Describe fully the single transformation that creates shape B from shape A.

Enlargement-Q2, IGCSE & GCSE Maths revision notes

We can see that the image is larger than the original object, therefore it must be an enlargement.
As the enlarged image is bigger than the original object, the scale factor must be greater than 1.
Compare two corresponding edges on the object and the image to find the scale factor.

The height of the original "H" is 3 squares
The height of the enlarged "H" is 9 squares

therefore Scale space Factor space equals 9 over 3 equals 3

Draw a straight line through the CoE and a pair of corresponding points on the original object and the enlarged image.
Repeat this step for as many vertices as you feel you need to so you can confidently locate the CoE.
Do this for all pairs of vertices to be sure!
The point of intersection of the lines is the CoE.

Enlargement-Q2-Working, IGCSE & GCSE Maths revision notes

Shape A has been enlarged using a scale factor of 3 and a centre of enlargement (9, 9) to create shape B

Fractional Enlargements

What is meant by fractional enlargements?

  • This is where the scale factor is less than 1
    • i.e. the scale factor is a fraction

How do I enlarge a shape with a fractional scale factor?

  • The process is no different as to when we are dealing with a positive scale factor
    • Measure the horizontal and vertical distances from the centre of enlargement (CoE) to any one vertex
    • Multiply these distances by the scale factor
      • This is where things will different as the 'new' distances will be smaller than the 'original' distances
    • Measure the new distances from the centre of enlargement to find the position of the new vertex
    • Repeat for all vertices or as many as you need to fill confident of being able to complete the enlargement
  • The big difference is in the visual result - the 'enlarged' shape will be smaller and closer to the centre of enlargement than the original
    • Mathematically though, these are still 'called enlargements'

How do I describe an enlargement with a fractional scale factor?

  • You need to be able to identify when a fractional scale factor has been using in an enlargement
    • Look for the enlarged shape being smaller than the original
  • To gain full marks you will still need to
    • state that the transformation is an enlargement
    • state the scale factor (as a fraction)
    • give the coordinates of the centre of enlargement

Examiner Tip

  • Still ensure you measure from the CoE when measuring distances to both the original shape and enlarged shape
  • Then check using lines through corresponding vertices, intersecting at the CoE still applies
    • However this can get confusing and untidy on small diagrams

Worked example

a)table row space row space end table
On the grid below enlarge shape C using scale factor 1 half and centre of enlargement (4, 2).

Write down the four vertices of your enlarged shape.cie-igcse-core-fractional-enlargements-rn-we-a1-image

Mark the centre of enlargement at (4, 2)
Count the number of squares horizontally and vertically to any vertex - we've chosen the vertex at (-2, 4)

cie-igcse-core-fractional-enlargements-rn-we-a2-image

Multiply these distance by the scale factor, 1 half

6 'right cross times 1 half equals 3 'right'

2 'down' cross times 1 half equals 1 'down'

Count these new distances (which should be smaller than the originals) from the CoE to find the corresponding point on the new image and mark it on

Repeat as required and draw lines through corresponding vertices and the CoE as a check

cie-igcse-core-fractional-enlargements-rn-we-a3-image

The four vertices of the enlarged shape are (0, 0), (0, 4),  (1, 3) and (1, 1)

Use a logical order, working your way round the shape slowly, to ensure you do miss any vertices out.

b)

Describe fully the single transformation that creates shape B from shape A.

cie-igcse-core-fractional-enlargements-rn-we-b1-image

We can see the image is smaller than the original so it is a fractional enlargement
Compare two corresponding edges to find the scale factor - we've used the top edge

scale factor = fraction numerator new space edge over denominator old space edge end fraction equals 2 over 6 equals 1 third

Draw straight lines through corresponding vertices on the original shape
Repeat this 3-4 times and you should find the lines intersect at the same point
This point will be the CoE

cie-igcse-core-fractional-enlargements-rn-we-b2-image

Shape A has been enlarged using a scale factor of bold 1 over bold 3 and a centre of enlargement (3, -3.5) to create shape B

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Negative Enlargements

How do I enlarge a shape if it has a negative scale factor?

  • You will still need to perform enlargements with negative scale factors
    • it is possible but unusual to be asked to identify one
  • Follow the same process as you would for a positive scale factor enlargement, the key things to look out for with a negative enlargement are:
    • the orientation of the object is changed as a negative enlargement rotates an object by 180o
    • when measuring the distance between the centre of enlargement (CoE) and the enlarged image, it is measured on the opposite side of the CoE

Examiner Tip

  • Remember to draw lines through the CoE and a vertex on the original object, this will remind you that the distances away from the CoE carry on in the opposite direction for a negative scale factor
  • Exam questions are quite keen on combining both negative and fractional scale factors, build your answer up following the rules and you will be fine!

Worked example

On the grid below enlarge shape F using scale factor  negative 1 third  and centre of enlargement  open parentheses 6 comma space minus 1 close parentheses. 

Label this shape F'.

If the area of F is 45 cm2 write down the area of F'.

ENSF-Q1, IGCSE & GCSE Maths revision notes

Start by marking the centre of enlargement (CoE) (6, -1) and selecting a starting vertex.

Count the horizontal and vertical distances from the vertex to the CoE.
Multiply those distances by the scale factor.

Vertex at (-4, 3)

Distance to CoE from vertex on original object: 3 to the right and 3 up

Distances from CoE to corresponding vertex on enlarged image: 3 cross times 1 third equals 1 to the right and 3 cross times 1 third equals 1 up

Counting the new distances from the CoE, on the other side from the original object, mark on the position of the corresponding point on the enlarged image.
Draw a straight line through the corresponding vertices and the CoE to check that they line up

Repeat this process for each vertex in turn.

ENSF-Q1-working, IGCSE & GCSE Maths revision notes

Connect the vertices as you go around so that you don't forget which should connect to which.

Remember, your enlarged image will be rotated by 180o.

ENSF-Q1-working-and-solution, IGCSE & GCSE Maths revision notes

The length scale factor is  1 third, meaning that each edge of the enlarged image is 1 third the length of the corresponding edge on the original object.

Find the area scale factor by squaring the length scale factor.

Area space scale space factor space equals open parentheses 1 third close parentheses squared equals 1 over 9

Multiply the area of the original object by the area scale factor to find the area of the enlarged image.

45 cross times 1 over 9

bold 5 bold space bold cm to the power of bold 2

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.