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Enlarging shapes

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visibility 173update 7 months agobookmarkshare

🎯 In this topic you will

  • Enlarge shapes using a positive whole number scale factor.
  • Enlarge shapes from a centre of enlargement.
  • Identify and describe enlargements.
  • Describe how the perimeter and area of squares and rectangles change when side lengths are enlarged.
 

🧠 Key Words

  • enlargement
  • scale factor
  • similar shapes
  • centre of enlargement
  • ray lines
Show Definitions
  • enlargement: A transformation that increases or decreases the size of a shape while keeping its proportions the same.
  • scale factor: The number that describes how much larger or smaller the image is compared to the original shape.
  • similar shapes: Shapes that have the same form and angles but may differ in size.
  • centre of enlargement: The fixed point from which distances are measured when enlarging a shape.
  • ray lines: Straight lines drawn from the centre of enlargement through points on the original shape to locate corresponding points on the image.
 

📐 Understanding Enlargement

An enlargement of a shape is a copy of the shape that changes the lengths, but keeps the same proportions.

Look at these two rectangles. The image is an enlargement of the object. Every length on the image is twice as long as the corresponding length on the object.

You say that the scale factor is $2$.

As the proportions are the same, the object and its image are called similar shapes.

In an enlargement all the angles stay the same size.

 

📘 Worked example

Draw the image of this triangle after an enlargement of scale factor 3.

Answer:

The height of the triangle is 1 square, so the height of the image is:

$3 \times 1 = 3$ squares

The base of the triangle is 2 squares, so the base of the image is:

$3 \times 2 = 6$ squares

Draw both these sides onto the grid.

Then draw the third side to complete the triangle.

When enlarging a shape, multiply each side length by the scale factor.

The shape keeps the same proportions, so angles remain unchanged.

Here, multiplying the base and height by 3 gives a new, similar triangle.

 

🧠 PROBLEM-SOLVING Strategy

Enlarging Shapes

Use these steps to enlarge shapes accurately on squared paper.

  1. Identify the scale factor of enlargement (e.g., $2$, $3$, $4$).
  2. Mark the centre of enlargement clearly with a dot.
  3. From the centre, draw rays through each vertex of the original shape.
  4. Multiply the distance from the centre to each vertex by the scale factor.
  5. Plot the new vertices at these enlarged positions.
  6. Join the vertices in order to form the enlarged shape.
  7. Check carefully:
    • All side lengths are multiplied by the scale factor.
    • All angles stay the same.
    • If the centre is inside the shape, the image will overlap the original.
    • If the centre is outside the shape, the image will be separate.
  8. For area and perimeter:
    • Perimeter is multiplied by the scale factor ($k$).
    • Area is multiplied by the scale factor squared ($k^2$).
  9. State the enlargement clearly: "Enlargement by scale factor $k$, centre at $(x, y)$."
Scale Factor Effect on Lengths Effect on Perimeter Effect on Area
$2$ Doubles ($\times 2$) Doubles ($\times 2$) Quadruples ($\times 4$)
$3$ Triples ($\times 3$) Triples ($\times 3$) Multiplies by $9$ ($3^2$)
$4$ Multiplies by $4$ Multiplies by $4$ Multiplies by $16$ ($4^2$)
 

❓ EXERCISES

1. Copy and complete these enlargements. Use a scale factor of $2$.

a.

👀 Show answer
The shape is enlarged by a scale factor of $2$, doubling each side length. The image should be twice as tall and twice as wide.

b. 

👀 Show answer
The shape is enlarged by a scale factor of $2$, doubling both horizontal and vertical lengths.

2. Copy each of these shapes onto squared paper. Enlarge each shape using its given scale factor.

a. Scale factor $2$

👀 Show answer
The enlarged shape is twice the size in both width and height, with corresponding sides doubled.

b. Scale factor $3$

👀 Show answer
The enlarged shape is three times the original size in both dimensions.

c. Scale factor $4$

👀 Show answer
The enlarged shape is four times the original size, keeping the same proportions.
 

🧠 Think like a Mathematician

Task: Enlarge the given polygon by a scale factor of $2$ and evaluate different methods.

Note: Only one side is horizontal and only one side is vertical; the other three sides are slanted across the grid.

Equipment: Squared paper, sharp pencil, ruler, compass (optional).

Method (try at least one of these):

  1. Ray-lines from a chosen centre. Pick a convenient centre (e.g., a vertex). Draw a straight ray from the centre through each vertex. Measure from the centre and mark each image point at $2$ times the original distance along its ray. Join the image points in order.
  2. Vector (coordinate) method. Choose an “origin” vertex $P$. Write the vectors to the other vertices (e.g., $\overrightarrow{PQ}=(\Delta x,\Delta y)$). Multiply each vector by $2$ to get image vectors, then plot $P'$ and $Q' = P' + 2\overrightarrow{PQ}$, etc.
  3. Grid counting with right-triangle scaffolds. For each slanted edge, form a right triangle showing its run and rise (e.g., “across 2, up 3”). Double both numbers (e.g., “across 4, up 6”) from the corresponding image vertex to rebuild the enlarged edges.

Follow-up Questions:

a. What methods could Inaya use to enlarge this shape?
b. Discuss the advantages and disadvantages of each method.
c. Which method is best here? Justify your choice.
Show Answers
  • a: (i) Ray-lines from a centre of enlargement, (ii) Vector/coordinate scaling from a chosen reference vertex, (iii) Grid counting using run–rise triangles on each edge.
  • b:
    • Ray-lines: + Works for any shape and scale factor; + visually clear; − requires careful use of a ruler and measuring distances from the centre; − slight drawing error can magnify.
    • Vector method: + Exact on coordinate grids; + easy arithmetic (multiply by $2$); − needs coordinates or accurate counting of $\Delta x,\Delta y$ for each vertex.
    • Grid run–rise: + Great for slanted edges; + no protractor needed; − can be slower (you must scaffold each edge) and relies on clean counting on the grid.
  • c: For this picture on squared paper, the vector/coordinate method is best: choose a convenient vertex as the centre/reference, keep that vertex fixed as $P'=P$, then double each vector to other vertices. It’s quick, reduces measurement error, and handles both horizontal/vertical and slanted sides consistently.
 

❓ EXERCISES

4. Copy each of these shapes onto squared paper. Enlarge each one, using its given scale factor.

👀 Show answer
a. The triangle is enlarged by a scale factor of $2$, so all side lengths are doubled.
b. The parallelogram is enlarged by a scale factor of $3$, so all side lengths are tripled.
c. The trapezium is enlarged by a scale factor of $4$, so all side lengths are multiplied by $4$.
 

❓ EXERCISES

5. Rafael has made a mistake. Explain the mistake he has made.

👀 Show answer
Rafael enlarged the side lengths by adding $3$ units instead of multiplying each distance from the centre of enlargement by $3$. His shape is not a true scale enlargement.

6. Draw the correct solution.

👀 Show answer
The correct enlargement is found by drawing rays from the centre of enlargement through each vertex of the original shape, and extending them so that each new vertex is $3$ times as far from the centre as the original. The enlarged shape should be $3$ times the size in all directions.
 

❓ EXERCISES

💡 Tip

Make sure you check that the scale factor works for every side of the shape.

7. The diagram shows a shape and its enlargement. What is the scale factor of the enlargement?

👀 Show answer
The larger shape is exactly $3$ times bigger than the smaller one in all corresponding sides. Therefore, the scale factor of the enlargement is $\mathbf{3}$.
 

❓ EXERCISES

8. Marcus and Sofia are looking at these flags. The flags are not drawn accurately.

a. Explain why flag C is not an enlargement of flag A.

👀 Show answer
For flag A, the base is $2\ \text{cm}$ and the height is $1\ \text{cm}$. For flag C, the base is $5\ \text{cm}$ and the height is $3\ \text{cm}$. The scale factor of the base is $\tfrac{5}{2} = 2.5$, while the scale factor of the height is $\tfrac{3}{1} = 3$. Since the scale factors are not the same, flag C is not an enlargement of flag A.

b. Explain why Sofia is correct.

👀 Show answer
For flag A, the base is $2\ \text{cm}$ and the height is $1\ \text{cm}$. For flag D, the base is $8\ \text{cm}$ and the height is $4\ \text{cm}$. Both the base and height are $4$ times larger, so the scale factor is $4$. Therefore, flag D is a true enlargement of flag A, which makes Sofia correct.
 

❓ EXERCISES

9. The diagram shows four rectangles: A, B, C and D.

a. Write down the scale factor of the enlargement of rectangles:

i. A to B

ii. A to C

iii. A to D

👀 Show answer
i. $2$
ii. $3$
iii. $4$

b. Work out the perimeter of rectangles:

i. A

ii. B

iii. C

iv. D

👀 Show answer
i. $P_A = 10\ \text{cm}$
ii. $P_B = 20\ \text{cm}$
iii. $P_C = 30\ \text{cm}$
iv. $P_D = 40\ \text{cm}$

c. Copy and complete this table. Write all the ratios in their simplest form.

Rectangles Scale factor of enlargement Ratio of lengths Ratio of perimeters
A : B $2$ $1 : 2$ $1 : 2$
A : C $3$ $1 : 3$ $1 : 3$
A : D $4$ $1 : 4$ $1 : 4$
👀 Show answer
Completed table shown above.

d. Write down a rule that connects the ratio of lengths to the ratio of perimeters.

👀 Show answer
The ratio of perimeters is the same as the ratio of lengths.

e. Will this rule work for any scale factor of enlargement? Will this rule work for any shape? Explain your answers.

👀 Show answer
Yes. The rule works for any scale factor of enlargement and for any shape, because perimeter is a linear measure, so it increases in direct proportion to the scale factor.
 

An enlargement of a shape is a copy of the shape that changes the lengths but keeps the same proportions. In an enlargement, all angles stay the same size.

Look at these two rectangles.

The image is an enlargement of the object.

Every length on the image is twice as long as the corresponding length on the object.

The scale factor is $2$.

The centre of enlargement tells you where to draw the image on a grid. In this case, as the scale factor is $2$, not only must the image be twice the size of the object, it must also be twice the distance from the centre of enlargement.

You can check you have drawn an enlargement correctly by drawing lines through the corresponding vertices of the object and image. All the lines should meet at the centre of the enlargement.

This is also a useful way to find the centre of enlargement if you are only given the object and the image.

 
📘 Worked example

Draw enlargements of the following triangles using the given scale factors and centres of enlargement, marked with a red dot.

a. scale factor $2$

b. scale factor $3$

Answer:

a. 

b. 

a. Start by looking at the corner of the triangle that is closest to the centre of enlargement (COE). This corner is 1 square to the right of the COE so, with a scale factor of $2$, the image will be 2 squares to the right of the COE.

Plot this point on the grid, then complete the triangle. Remember to double all the lengths.

b. One of the corners of this triangle is on the centre of enlargement, so this corner doesn’t move.

Look at the bottom right corner of the triangle. This corner is 1 square to the right and 1 square down from the COE. With a scale factor of $3$, the image will be 3 squares to the right and 3 squares down from the COE.

Plot this point on the grid, then complete the triangle. Remember to multiply all the other lengths by $3$.

 

🧠 PROBLEM-SOLVING Strategy

Enlarging Shapes with a Scale Factor

Use these steps to enlarge any polygon on a coordinate grid, given a scale factor and a centre of enlargement.

  1. Mark the centre of enlargement at $(x_c, y_c)$.
  2. For each vertex $(x, y)$ of the original shape:
    • Find the displacement from the centre: $(x - x_c, \; y - y_c)$.
    • Multiply this displacement by the scale factor$k$.
    • Add back the centre coordinates to get the new vertex: $(x', y') = (x_c, y_c) + k \times (x - x_c, \; y - y_c)$.
  3. Repeat for all vertices and join them in order to complete the image.
  4. If $k > 1$, the shape enlarges; if $0 < k < 1$, it reduces.
  5. Check: draw ray lines through original and image vertices — they should meet at the centre.
  6. Remember:
    • Perimeter is multiplied by $k$.
    • Area is multiplied by $k^2$.
Original Vertex Centre $(1,1)$ Scale Factor $k=3$ Image Vertex
$(2,2)$ $(1,1)$ $3$ $(4,4)$
$(4,2)$ $(1,1)$ $3$ $(10,4)$
 

❓ EXERCISES

1. Copy each shape onto squared paper.
Enlarge each shape using the given scale factor and centre of enlargement.

a. scale factor $2$

b. scale factor $3$

c. scale factor $4$

d. scale factor $2$

e. scale factor $3$

f. scale factor $4$

👀 Show answer
Each shape should be enlarged correctly on squared paper, keeping the centre of enlargement fixed and scaling distances by the given scale factor.

💡 Tip

Make sure you leave enough space around your shape to complete the enlargement.

2. This is part of Geraint’s homework.

a. Explain Geraint’s mistake.

b. Make a copy of the triangle on squared paper.
Draw the correct enlargement.

👀 Show answer

a. Geraint enlarged the triangle but did not measure distances from the correct centre of enlargement. The vertices were placed incorrectly relative to the red dot.

b. The correct enlargement should be drawn by doubling all distances from the centre of enlargement, ensuring all points are placed consistently.

 

❓ EXERCISES

3. The vertices of this triangle are at $(2,2)$, $(2,3)$ and $(4,2)$.

a. Copy the diagram on squared paper.
Mark with a dot the centre of enlargement at $(1,1)$.
Enlarge the triangle with scale factor $3$ from the centre of enlargement.

b. Write the coordinates of the vertices of the image.

👀 Show answer

a. The enlarged triangle should be drawn by multiplying all distances from the centre of enlargement $(1,1)$ by $3$.

b. New coordinates:
From $(1,1)$ to $(2,2)$ is $(+1,+1)$. Enlarged ×$3$ → $(+3,+3)$ → image point $(4,4)$.
From $(1,1)$ to $(2,3)$ is $(+1,+2)$. Enlarged ×$3$ → $(+3,+6)$ → image point $(4,7)$.
From $(1,1)$ to $(4,2)$ is $(+3,+1)$. Enlarged ×$3$ → $(+9,+3)$ → image point $(10,4)$.

So the coordinates of the image are $(4,4)$, $(4,7)$, and $(10,4)$.

 

🧠 Think like a Mathematician

4. Marcus and Arun enlarge this square using scale factor $3$.
Marcus uses a centre of enlargement at $(1,1)$.
Arun uses a centre of enlargement at $(0,1)$.
Read what Marcus and Arun say.

Marcus: “There is one invariant point on my object and image.”
Arun: “There are no invariant points on my object and image.”

a. Make two copies of the grid above and enlarge the square using scale factor $3$ with:
i. Marcus’s centre of enlargement
ii. Arun’s centre of enlargement

b. Look at the diagrams you draw for part a.
What do you think Marcus and Arun mean by ‘invariant points’?

c. Describe where a centre of enlargement must be, for you to have one invariant point.

d. Describe where a centre of enlargement must be, for you to have no invariant points.
Reflect on your answers independently.

👀 Show Answer
  • a.i. Enlarging about $(1,1)$ with scale factor $3$ keeps the point $(1,1)$ fixed as an invariant point. The square expands away from this centre.
  • a.ii. Enlarging about $(0,1)$ with scale factor $3$ means no vertex coincides with its image. There are no invariant points.
  • b. An invariant point is one that maps onto itself after enlargement. Marcus has one invariant point (the centre of enlargement lies on the shape), while Arun has none.
  • c. For there to be one invariant point, the centre of enlargement must lie on the shape.
  • d. For there to be no invariant points, the centre of enlargement must lie outside the shape.
 

❓ EXERCISES

5. The vertices of this trapezium are at $(3,2)$, $(7,2)$, $(5,4)$ and $(4,4)$.

a. Copy the diagram onto squared paper.
Mark with a dot the centre of enlargement at $(5,2)$.
Enlarge the trapezium with scale factor $2$ from the centre of enlargement.

b. Write the coordinates of the vertices of the image.

c. Write the coordinates of the invariant point.

👀 Show answer

a. Enlarge each vertex relative to the centre of enlargement $(5,2)$ by scale factor $2$.

b. Work out each new coordinate:

  • From $(5,2)$ to $(3,2)$ is $(-2,0)$. Doubled → $(-4,0)$. Image point: $(1,2)$.
  • From $(5,2)$ to $(7,2)$ is $(+2,0)$. Doubled → $(+4,0)$. Image point: $(9,2)$.
  • From $(5,2)$ to $(5,4)$ is $(0,+2)$. Doubled → $(0,+4)$. Image point: $(5,6)$.
  • From $(5,2)$ to $(4,4)$ is $(-1,+2)$. Doubled → $(-2,+4)$. Image point: $(3,6)$.

So the coordinates of the image are: $(1,2)$, $(9,2)$, $(5,6)$, $(3,6)$.

c. The invariant point is the centre of enlargement itself: $(5,2)$.

 

❓ EXERCISES

6. Each diagram shows an object and its image after an enlargement.
For each part, write down the scale factor of the enlargement.

a. 

b. 

👀 Show answer

a. The smaller base is about $3$ units long, and the enlarged base is about $6$ units long. So the scale factor is $2$.

b. The small vertical side is about $2$ units, and the enlarged side is about $6$ units. So the scale factor is $3$.

 

❓ EXERCISES

7. The diagram shows shape $ABCD$ and its image $A'B'C'D'$.

a. Write the scale factor of the enlargement.
Read what Marcus and Zara say:

Marcus: “I think the centre of enlargement is at $(-3,-4)$.”

Zara: “I think the centre of enlargement is at $(-4,-3)$.”

b. Who is correct? Explain how you worked out your answer.

👀 Show answer

a. The enlargement doubles the lengths, so the scale factor is $2$.

b. Marcus is correct. If you draw lines through corresponding vertices of the object and the image, they meet at $(-3,-4)$. That is the true centre of enlargement, not $(-4,-3)$.

 

🧠 Think like a Mathematician

8. Zara drew a triangle with vertices at $(1,1)$, $(2,1)$ and $(1,3)$. She enlarged the shape by a scale factor of $3$, centre $(0,0)$. Read what Zara said:

Zara: “If I multiply the coordinates of each vertex by $3$ it will give me the coordinates of the enlarged triangle, which are at $(3,3)$, $(6,3)$, and $(3,9)$.”

a. Show, by drawing, that in this case Zara is correct.
Read what Arun said:

Arun: “This means that, for any enlargement, with any scale factor and centre of enlargement, I can multiply the coordinates of each vertex by the scale factor to work out the coordinates of the enlarged shape.”

b. Use a counter-example to show that Arun is wrong.

c. What are the only coordinates of the centre of an enlargement where you can multiply the coordinates of the vertices of the object to get the coordinates of the vertices of the image?

💡 Tip

A counter-example is just one example that shows a statement is not true.

👀 Show Answer
  • a. Drawing the enlargement confirms Zara’s method works when the centre of enlargement is $(0,0)$. The new vertices are at $(3,3)$, $(6,3)$, $(3,9)$.
  • b. A counter-example: Take the triangle $(1,1)$, $(2,1)$, $(1,3)$ with centre of enlargement at $(1,1)$ and scale factor $2$. Multiplying all coordinates by $2$ would not give the correct image, because the centre is not at the origin.
  • c. The only case where you can multiply all coordinates by the scale factor directly is when the centre of enlargement is at the origin $(0,0)$.
 

🔍 Properties of Enlargement

You already know that when you enlarge a shape:

  • all the lengths of the sides of the shape increase in the same proportion
  • all the angles in the shape stay the same size.

📘 Enlargement with Different Centres

You enlarged shapes using a centre of enlargement outside or on the shape. In this section you will enlarge shapes using a centre of enlargement inside the shape. You will also look at the effect of an enlargement on the perimeter and area of squares and rectangles.

✏️ Describing Enlargements

Remember that when you describe an enlargement you must give:

  • the scale factor of the enlargement
  • the position of the centre of enlargement.
 
📘 Worked example

a. The diagram shows a trapezium.
Draw an enlargement of the trapezium, with scale factor $2$, centre of enlargement shown.

b. The diagram shows two triangles, A and B.
Triangle B is an enlargement of triangle A.
Describe the enlargement.

Answer:

a.

The closest vertex of the trapezium is one square left and one square down from the centre of enlargement. On the enlarged trapezium, this vertex will be two squares left and two squares down from the centre of enlargement. Mark this vertex on the diagram, then complete the trapezium. The length of each side is twice the length of that side in the original shape.

b.

First, work out the scale factor of the enlargement. Compare matching sides of the triangles – for example, the two sides marked with red arrows. In triangle A, the length is $2$ squares and in triangle B the length is $4$ squares.

$4 \div 2 = 2$, so the scale factor is $2$.

Now find the centre of enlargement by drawing ray lines through the corresponding vertices of the triangles, shown by the purple lines. The purple lines meet at $(4,3)$.

The enlargement has scale factor $2$, centre $(4,3)$.

To enlarge a shape, multiply all distances from the centre of enlargement by the scale factor.

Here, the trapezium is enlarged by scale factor $2$, so each distance doubles. For the triangles, comparing side lengths gives the scale factor, and drawing ray lines locates the centre of enlargement.

 

🧠 PROBLEM-SOLVING Strategy

Enlarging Shapes

Follow these steps to enlarge a shape using a given scale factor and centre of enlargement.

  1. Identify the scale factor$k$ and the centre of enlargement$(x_c, y_c)$.
  2. For each vertex $(x,y)$ of the original shape, measure its displacement from the centre: $(x - x_c, y - y_c)$.
  3. Multiply this displacement by the scale factor $k$ to get the new displacement.
  4. Add this new displacement back to the centre to find the image vertex: $(x',y') = (x_c, y_c) + k \times (x - x_c, y - y_c)$.
  5. Repeat for all vertices, then join them in order to complete the enlarged shape.
  6. If $k > 1$, the image is larger and further from the centre; if $0 < k < 1$, it is smaller and closer.
  7. The perimeter of the shape is multiplied by $k$, and the area is multiplied by $k^2$.
  8. Check correctness by drawing ray lines through corresponding vertices — they should all meet at the centre of enlargement.
 

❓ EXERCISES

1. Copy and complete this enlargement with scale factor $3$ and centre of enlargement shown.

👀 Show answer
The shape should be enlarged by scale factor $3$ with respect to the red dot (centre of enlargement). Each vertex of the image is placed three times as far from the centre of enlargement along the same line as the corresponding vertex of the object. The completed figure is the correctly enlarged shape.
 

❓ EXERCISES

💡 Tip

Make sure you leave enough space around your shape to complete the enlargement.

2. Copy each of these shapes on to squared paper.
Enlarge each shape using the given scale factor and centre of enlargement shown.

a. scale factor $2$

b. scale factor $3$

c. scale factor $4$

👀 Show answer

Each shape should be enlarged correctly using the given scale factor and the marked centre of enlargement. All sides scale proportionally, and all angles remain unchanged.

3. This is part of Tasha’s homework.

a. Explain the mistake Tasha has made.

b. Make a copy of the triangle on squared paper.
Draw the correct enlargement.

👀 Show answer

a. Tasha’s mistake was that she did not measure distances from the centre of enlargement. She enlarged the triangle but did not keep it aligned with the red centre point.

b. The correct enlargement must be drawn so that each vertex is exactly twice as far from the centre of enlargement along the same line as the original vertex.

 

❓ EXERCISES

4. The diagram shows square A on a coordinate grid.
Make three copies of the diagram on squared paper.

a. On the first copy, draw an enlargement of the shape with scale factor $2$, centre $(7,5)$. Label the image B.

b. On the second copy, draw an enlargement of the shape with scale factor $3$, centre $(5,6)$. Label the image C.

c. On the third copy, draw an enlargement of the shape with scale factor $4$, centre $(5,5)$. Label the image D.

👀 Show answer

a. The enlargement has scale factor $2$, centre $(7,5)$. Image B is twice as far from the centre as the original square A.

b. The enlargement has scale factor $3$, centre $(5,6)$. Image C is three times as far from the centre as the original square A.

c. The enlargement has scale factor $4$, centre $(5,5)$. Image D is four times as far from the centre as the original square A.

In each case, all side lengths scale in proportion and angles remain unchanged.

 

🧠 Think like a Mathematician

5. Work independently to answer these questions.

a. Look back at your diagrams in Question 4.

i. Work out the perimeter of each square A, B, C and D.
ii. Work out the area of each square A, B, C and D.

b. Copy and complete this table. Write all the ratios in their simplest form.

Squares Scale factor of enlargement Ratio of lengths Ratio of perimeters Ratio of areas
A : B 2 1 : 2 1 : 2 1 : 4
A : C 3 1 : 3 1 : 3 1 : 9
A : D 4 1 : 4 1 : 4 1 : 16

c. Write a rule that connects the ratio of lengths to the ratio of perimeters.

d. Write a rule that connects the ratio of lengths to the ratio of areas.

e. Will these rules work for any scale factor of enlargement? Will these rules work for any shape?

f. Compare your answers with other learners. Discuss any differences.

💡 Tip

In part d, remember the square numbers: $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, etc.

👀 show answer
  • a.i. Perimeters scale directly with the enlargement. B is twice A, C is three times A, D is four times A.
  • a.ii. Areas scale with the square of the scale factor. B is four times A, C is nine times A, D is sixteen times A.
  • c. The ratio of perimeters equals the ratio of lengths.
  • d. The ratio of areas equals the square of the ratio of lengths.
  • e. Yes, these rules work for any scale factor and for any shape, since enlargements preserve similarity.
  • f. Answers may differ in wording, but should all reflect the proportional relationship between lengths, perimeters, and areas.
 

❓ EXERCISES

6. A rectangle, R, has a perimeter of $14\ \text{cm}$ and an area of $10\ \text{cm}^2$.
The rectangle is enlarged by a scale factor of $3$ to become rectangle T.
Copy and complete the workings to find the perimeter and area of rectangle T.

Perimeter of R = $14\ \text{cm}$ → Perimeter of T = $14 \times 3 =$ ____ $\text{cm}$

Area of R = $10\ \text{cm}^2$ → Area of T = $10 \times 3^2 =$ ____ $\text{cm}^2$

👀 Show answer
Perimeter of T = $14 \times 3 = 42\ \text{cm}$
Area of T = $10 \times 3^2 = 10 \times 9 = 90\ \text{cm}^2$

7. A triangle, G, has a perimeter of $12\ \text{cm}$ and an area of $6\ \text{cm}^2$.
The triangle is enlarged by a scale factor of $5$ to become triangle H.
Work out the perimeter and area of triangle H.

👀 Show answer
Perimeter of H = $12 \times 5 = 60\ \text{cm}$
Area of H = $6 \times 5^2 = 6 \times 25 = 150\ \text{cm}^2$

💡 Tip

Some ray lines have been drawn on the diagram to help you.

8. The diagram shows shapes F and G on a square grid.
Copy and complete this statement:
Shape G is an enlargement of shape F, scale factor ____ and centre of enlargement $(\ \ ,\ \ )$.

👀 Show answer
Shape G is an enlargement of shape F with scale factor $3$ and centre of enlargement $(1,2)$.
 

❓ EXERCISES

💡 Tip

Remember: to describe an enlargement, you need to write ‘Enlargement’ and give the scale factor and the centre of enlargement.

9. The diagram shows three triangles, A, B and C, on a coordinate grid.
a. Triangle B is an enlargement of triangle A. Describe the enlargement.
b. Triangle C is an enlargement of triangle A. Describe the enlargement.

👀 Show answer

a. Triangle B is an enlargement of triangle A with scale factor $\tfrac{1}{2}$ and centre of enlargement at $(0,0)$.

b. Triangle C is an enlargement of triangle A with scale factor $2$ and centre of enlargement at $(0,0)$.

10. The vertices of rectangle X are at $(1,-2)$, $(1,-3)$, $(3,-3)$ and $(3,-2)$.
The vertices of rectangle Y are at $(-5,4)$, $(-5,1)$, $(1,1)$ and $(1,4)$.
Rectangle Y is an enlargement of rectangle X.
Describe the enlargement.

👀 Show answer
Rectangle Y is an enlargement of rectangle X with scale factor $3$ and centre of enlargement at $(1,-3)$.
 

🧠 Think like a Mathematician

11. Work independently to answer this question.

Arun makes this conjecture:

“When one shape is an enlargement of another, and the centre of enlargement is inside the shapes, I don’t think you can use ray lines to find the centre of enlargement.”

Task: Do you agree or disagree with Arun’s conjecture? Show working to justify your answer.

👀 Show Answer

Arun’s conjecture is not correct. You can always use ray lines (lines drawn through corresponding vertices of the object and image) to locate the centre of enlargement. Even if the centre lies inside the shape, extending the rays far enough will show where they intersect. That intersection is the centre of enlargement.

This works because enlargements preserve straight-line relationships, so all corresponding vertices are collinear with the centre of enlargement, regardless of its position inside or outside the shape.

 

❓ EXERCISES

12. The vertices of shape K are at $(4,7)$, $(7,7)$, $(7,4)$ and $(5,5)$.
The vertices of shape L are at $(0,11)$, $(9,11)$, $(9,2)$ and $(3,5)$.
Shape L is an enlargement of shape K. Describe the enlargement.

👀 Show answer
Enlargement with scale factor $3$ and centre of enlargement $(6,5)$.
Check: Using $(x',y')=(6,5)+3\big((x,y)-(6,5)\big)$ maps K’s vertices to L’s vertices.
 

❓ EXERCISES

13. The diagram shows two shapes, A and B. Shape B is an enlargement of shape A. Describe the enlargement.

👀 Show answer
Shape B is an enlargement of shape A with scale factor $2$ and centre of enlargement at $(1,2)$.
 

📘 What we've learned

  • An enlargement changes the size of a shape but preserves its angles and proportions.
  • To describe an enlargement, we must state both the scale factor and the centre of enlargement.
  • The scale factor tells how many times bigger or smaller the shape becomes:
    • Side lengths are multiplied by the scale factor.
    • Perimeter also multiplies by the same scale factor.
    • Area multiplies by the square of the scale factor $(k^2)$.
  • If the centre of enlargement is at the origin $(0,0)$, the image coordinates can be found by multiplying each coordinate by the scale factor.
  • An invariant point is a point that maps onto itself during enlargement (always the centre of enlargement).
  • Ray lines drawn through corresponding points always meet at the centre of enlargement.

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