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
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

Exam 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.

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.

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'.

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 2 ²= 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.

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.

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

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

Exam 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 cm ² write down the area of F'.

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.

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.

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

$5 {cm}^{2}$

1. Number

Number Toolkit

Mathematical Operations
Negative Numbers
Money Calculations
Number Operations
Related Calculations
Counting Principles

Prime Factors, HCF & LCM

Types of Number
Prime Factor Decomposition
HCF & LCM

Powers, Roots & Standard Form

Powers, Roots & Indices
Standard Form

Fractions

Basic Fractions
Operations with Fractions

Percentages

Basic Percentages
Working with Percentages

Simple & Compound Interest, Growth & Decay

Interest & Depreciation
Exponential Growth & Decay

Fractions, Decimals & Percentages

Converting between FDP
Converting between FDP

Rounding, Estimation & Bounds

Rounding & Estimation
Bounds

Surds

Simplifying Surds
Rationalising Denominators

Using a Calculator

Using a Calculator

2. Algebra

Algebra Toolkit

Algebraic Notation & Vocabulary
Algebra Basics

Algebraic Roots & Indices

Algebraic Roots & Indices

Expanding Brackets

Expanding Single Brackets
Expanding Multiple Brackets

Factorising

Factorising
Factorising Quadratics
Quadratics Factorising Methods

Completing the Square

Completing the Square

Rearranging Formulae

Rearranging Formulae

Algebraic Proof

Algebraic Proof

Linear Equations

Solving Linear Equations

Solving Quadratic Equations

Solving Quadratic Equations
Quadratic Equation Methods

Simultaneous Equationsr

Simultaneous Equations

Iteration

Iteration

Forming & Solving Equations

Forming Equations
Equations & Problem Solving

Functions

Functions Toolkit
Composite & Inverse Functions

Coordinate Geometrys

Coordinates
Coordinate Geometry

Linear Graphs y = mx + c

Straight Line Graphs (y = mx + c)
Parallel & Perpendicular Lines

Graphs of Functions

Types of Graphs
Graphical Solutions
Trig Graphs

Equation of a Circle

Equation of a Circle
Equation of a Tangents

Estimating Gradients & Areas under Graphs

Finding Gradients of Tangents
Finding Areas under Graphs

Real-Life Graphs

Distance-Time & Speed-Time Graphs
Conversion Graphs
Rates of Change of Graphs

Solving Inequalities

Solving Linear Inequalities
Conversion Graphs
Solving Quadratic Inequalities

Graphing Inequalities

Graphing Inequalities

Transformations of Graphs

Reflections of Graphs

Sequences

Introduction to Sequences
Types of Sequences
Linear Sequences
Quadratic Sequences

3. Ratio, Proportion & Rates of Change

Ratio Toolkit

Simple Ratio
Working with Proportion

Ratio Problem Solving

Ratios & FDP
Multiple Ratios

Direct & Inverse Proportions

Direct & Inverse Proportion

Standard & Compound Units

Time
Unit Conversions
Compound Measures

Exchange Rates & Best Buys

Exchange Rates & Best Buys

4. Geometry & Measures

Geometry Toolkit

Symmetry
2D & 3D Shapes
Plans & Elevations

Angles in Polygons & Parallel Lines

Basic Angle Properties
Angles in Polygons
Angles in Parallel Lines

Bearings, Scale Drawing, Constructions & Loci

Bearings
Scale & Maps
Constructing Triangles
Constructions & Loci

Circle Theorems

Angles at Centre & Semicircles
Chords & Tangents
Cyclic Quadrilaterals
Segment Theorems
Circle Theorem Proofs

Area & Perimeter

Area & Perimeter
Problem Solving with Areas

Circles, Arcs & Sectors

Area & Circumference of Circles
Arcs & Sectors

Volume & Surface Area

Volume
Surface Area

Congruence, Similarity & Geometrical Proof

Congruence
Similarity
Geometrical Proof

Area & Volume of Similar Shapes

Similar Area & Volumes

Right-Angled Triangles – Pythagoras & Trigonometry

Pythagoras Theorem
Right-Angled Trigonometry
Exact Trig Values

Sine, Cosine Rule & Area of Triangles

Sine & Cosine Rules
Area of a Triangle
Applications of Trigonomet

3D Pythagoras & Trigonometry

3D Pythagoras & Trigonometry

Vectors

Introduction to Vectors
Working with Vectors

Transformations

Translations
Reflections
Rotations
Enlargements
Combination of Transformations

5. Probability

Probability Toolkit

Basic Probability
Relative & Expected Frequency

Simple Probability Diagrams

Two Way Tables
Frequency Trees
Set Notation & Venn Diagrams

Tree Diagrams

Tree Diagrams

Combined & Conditional Probability

Combined Probability
Conditional Probability
Combined Conditional Probabilities

6. Statistics

Statistics Toolkit

Mean, Median & Mode
Averages from Tables
Range & Quartiles
Comparing Distributions
Population & Sampling

Statistical Diagrams

Bar Charts & Pictograms
Pie Charts
Time Series Graphs
Working with Statistical Diagrams

Histograms

Histograms

Cumulative Frequency & Box Plots

Cumulative Frequency Diagrams
Box Plots

Scatter Graphs & Correlation

Scatter Graphs

Top Rated by Parents/Students Nationwide

Enlargements

Enlargements

What is an enlargement?

How do I enlarge a shape?

How do I describe an enlargement?

Which points are invariant with an enlargement?

Exam Tip

Worked example

Negative Enlargements

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

Exam Tip

Worked example

Leading tutoring agency within the North East, that specialises in getting students top grades all over the country.

Quick Links

Social

Get In Touch

Get Started

Book A Demo