Enlargements

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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
The position of a shape will also change with enlargement
The orientation of the shape will be the same for a positive enlargement

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
one vertex on the original object
STEP 2:
Multiply the distances by the given scale factor
STEP 3:
Starting again from the centre of enlargement, measure the new distances and mark the position on the grid of the corresponding vertex on the enlarged image
- The distances from the centre of enlargement to the enlarged image will be in the same direction for a positive scale factor and the opposite direction for a negative scale factor
STEP 4:
Repeat STEPs 1 to 3 for the remaining vertices
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

Which points are invariant with an enlargement?

Invariant points are points that do not change position when a transformation has been performed
- Invariant points don't move!
With an enlargement, if the centre of enlargement is a point on the object, then that point is invariant
- if the centre of enlargement is not a point on the object, then there are no invariant points

Examiner Tip

Make sure that you always start from the centre of enlargement when measuring distances to the original object and 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

Worked example

(a)

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

Label your translated shape C'.

Without working out any areas explain why the area of C' is four times as large as the area of 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.
Join adjacent vertices on the enlarged image as you go.
Label the enlarged image C'.

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

Because the scale factor is 2, each of the lengths will be twice as long on the enlarged image as they are on the original object.
Square the length scale factor to find the area scale factor.

The length scale factor is 2, therefore the area scale factor will be 2²= 4, hence, the area of C' is 4 times larger than C

(b)

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

Enlargement-Q2, IGCSE & GCSE Maths revision notes

You 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.

$Scale Factor = \frac{9}{3} = 3$

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 all of the pairs of vertices on the object and image.
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

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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

The key things with a negative enlargement are:
- the orientation of the object is changed as a negative enlargement rotates an object by 180^o
- 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 $- \frac{1}{3}$ and centre of enlargement $(6, - 1)$ .

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: $3 \times \frac{1}{3} = 1$ to the right and $3 \times \frac{1}{3} = 1$ 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 180^o.

The length scale factor is $\frac{1}{3}$ , meaning that each edge of the enlarged image is $\frac{1}{3}$ the length of the corresponding edge on the original object.

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

$Area scale factor = {(\frac{1}{3})}^{2} = \frac{1}{9}$

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

$45 \times \frac{1}{9}$

$5 {cm}^{2}$

Enlargements (AQA GCSE Maths)

Revision Note

Enlargements

What is an enlargement?

How do I enlarge a shape?

How do I describe an enlargement?

Which points are invariant with an enlargement?

Examiner Tip

Worked example

Negative Enlargements

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

Examiner Tip

Worked example

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Author: Naomi C