Chapter 3 of 8

All 8 chapters
  1. 1Reflections and rotations, readunderstood
  2. 2Translations and combined moves, readunderstood
  3. 3Enlargements, readunderstood
  4. 4Congruent triangles, readunderstood
  5. 5Similar shapes, readunderstood
  6. 6Similar areas and volumes, readunderstood
  7. 7Arcs and sectors, readunderstood
  8. 8Angle rules in circles, readunderstood

M4.4.3 Chapter 3

Enlargements

A shape is enlarged by scale factor 2 about a point. Where does a corner that is 3 squares right of the point go?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Enlarge a shape by a scale factor from a centre
  • Enlarge a shape by a scale factor of 1/2
  • Find the centre of an enlargement

An enlargement changes the size of a shape and keeps its form. To describe one you need two things. The first is a point called the centre of enlargement. The second is a number called the scale factor. Every distance from the centre is multiplied by the scale factor.

Draw it first

Enlarge triangle A by scale factor 2 about C

  1. A grid with an x axis and a y axis from 0 to 8. Step 1: triangle A with corners at (2, 2), (4, 2) and (2, 3), labelled A, and the centre C at (1, 1). Step 2: a dashed line from C through each corner of A, ending at the image corners (3, 3), (7, 3) and (3, 5). Step 3: the image, triangle A′, with corners at (3, 3), (7, 3) and (3, 5), labelled A′.12345678123456780xyCA

    1 Triangle A has corners at (2, 2), (4, 2) and (2, 3). The centre of enlargement is C at (1, 1).

  2. A grid with an x axis and a y axis from 0 to 8. Step 1: triangle A with corners at (2, 2), (4, 2) and (2, 3), labelled A, and the centre C at (1, 1). Step 2: a dashed line from C through each corner of A, ending at the image corners (3, 3), (7, 3) and (3, 5). Step 3: the image, triangle A′, with corners at (3, 3), (7, 3) and (3, 5), labelled A′.12345678123456780xyCA

    2 Draw a line from C through each corner of A. Go on to twice the distance from C. Mark each end.

  3. A grid with an x axis and a y axis from 0 to 8. Step 1: triangle A with corners at (2, 2), (4, 2) and (2, 3), labelled A, and the centre C at (1, 1). Step 2: a dashed line from C through each corner of A, ending at the image corners (3, 3), (7, 3) and (3, 5). Step 3: the image, triangle A′, with corners at (3, 3), (7, 3) and (3, 5), labelled A′.12345678123456780xyCAA′

    3 Join the three marks. This is the image, A′. Each side is twice as long as the side it matches.

Look at the corner (2, 2). It is 1 right and 1 up from C. Its image is 2 right and 2 up from C, which is the point (3, 3). The rule is: image = C + scale factor × (corner − C).

An enlargement changes lengths and keeps angles. Each side of A′ is twice the side it matches. Every angle of A′ equals the angle it matches. That is why the shape keeps its form.

A grid with an x axis and a y axis from 0 to 8. The centre C at (1, 1). Triangle A with corners at (2, 2), (4, 2) and (2, 3), labelled A, and triangle A′ with corners at (3, 3), (7, 3) and (3, 5), labelled A′. Dashed lines run from C through the corners of A to the corners of A′.12345678123456780xyCAA′
The image corners lie on the lines from C, at twice the distance.

Worked example

Enlarge the triangle with corners (2, 2), (4, 2) and (2, 3) by scale factor 2 about C at (1, 1).

  1. Find each corner as a step from C. The corner (2, 2) is 1 right and 1 up. The corner (4, 2) is 3 right and 1 up. The corner (2, 3) is 1 right and 2 up.
  2. Multiply each step by 2. The steps become 2 right and 2 up, then 6 right and 2 up, then 2 right and 4 up.
  3. Count from C at (1, 1). The image corners are (3, 3), (7, 3) and (3, 5).

Answer: (3, 3), (7, 3) and (3, 5)

A scale factor between 0 and 1 makes the shape smaller. With scale factor 1/2 every distance from the centre is halved. The image is half as long and lies between the centre and the original.

A grid with an x axis and a y axis from 0 to 8. The centre C at (1, 1). Triangle A with corners at (3, 3), (7, 3) and (3, 7), labelled A, and the smaller triangle A′ with corners at (2, 2), (4, 2) and (2, 4), labelled A′. Dashed lines run from C through the corners of A′ to the corners of A.12345678123456780xyCAA′
Scale factor 1/2 about C. Triangle A′ is half the size of A.

Worked example

Enlarge the triangle with corners (3, 3), (7, 3) and (3, 7) by scale factor 1/2 about C at (1, 1).

  1. Find each corner as a step from C. The corner (3, 3) is 2 right and 2 up. The corner (7, 3) is 6 right and 2 up. The corner (3, 7) is 2 right and 6 up.
  2. Halve each step. The steps become 1 right and 1 up, then 3 right and 1 up, then 1 right and 3 up.
  3. Count from C at (1, 1). The image corners are (2, 2), (4, 2) and (2, 4). Each side is half as long.

Answer: (2, 2), (4, 2) and (2, 4)

Now suppose you are given a shape and its image, and you need the centre. The line through a corner and its image passes through the centre. Draw two such lines. They meet at the centre.

A grid with an x axis and a y axis from 0 to 8. Triangle A with corners at (2, 3), (4, 3) and (2, 4), labelled A, and the larger triangle A′ with corners at (3, 4), (7, 4) and (3, 6), labelled A′. Dashed lines run from each corner of A′ through the matching corner of A to the point C at (1, 2).12345678123456780xyCAA′
The lines through matching corners all meet at C.

Worked example

Triangle A has corners (2, 3), (4, 3) and (2, 4). Its image has corners (3, 4), (7, 4) and (3, 6). Find the centre and the scale factor.

  1. Join the corner (2, 3) to its image (3, 4). Draw the line on past (2, 3).
  2. Do the same for the corner (4, 3) and its image (7, 4).
  3. The two lines cross at (1, 2). That point is the centre.
  4. From C the corner (2, 3) is 1 across. The image (3, 4) is 2 across. The scale factor is 2.

Answer: centre (1, 2), scale factor 2

See it three ways

With things

Draw a small flag on squared paper. Draw it again on paper with squares 3 times as wide. Every length is 3 times as long.

As a picture

A grid with an x axis and a y axis from 0 to 8. The centre C at (1, 1). Triangle A with corners at (2, 2), (3, 2) and (2, 3), labelled A, and the larger triangle A′ with corners at (4, 4), (7, 4) and (4, 7), labelled A′. Dashed lines run from C through the corners.12345678123456780xyCAA′

In symbols

With scale factor 3 about C at (1, 1), the corner (3, 2) goes to (7, 4). Every length is 3 times as long.

Try it

Double a triangle with a ruler

You need: squared paper, a ruler, a pencil.

  1. Mark a point C near the bottom left of the paper. Draw a small triangle some way from C.
  2. Draw a line from C through each corner of the triangle. Make each line long.
  3. Measure from C to a corner. Mark a point on the line that is twice that distance from C. Do the same for each corner.
  4. Join the three new points. Measure one side of each triangle.

What you will see: Each side of the new triangle is twice the side it matches. Check the angles with a corner of the paper: they look the same.

If you cannot try it, this is what happens.
A grid with an x axis and a y axis from 0 to 8. The centre C at (1, 1). Triangle A with corners at (3, 2), (4, 2) and (3, 4), labelled A, and the larger triangle A′ with corners at (5, 3), (7, 3) and (5, 7), labelled A′. Dashed lines run from C through the corners.12345678123456780xyCAA′

A wrong idea many people have

To enlarge a shape by scale factor 2, add 2 to every length.

Why it feels right: People say "make it 2 bigger", and adding is the first way to make a number bigger.

The test that breaks it: Take a triangle with sides 3 cm and 6 cm. Add 2 to each and they become 5 cm and 8 cm. The first shape had a 1 : 2 ratio of sides. The new one does not, so it is a different shape.

What is true: An enlargement multiplies every length by the scale factor. The sides 3 cm and 6 cm become 6 cm and 12 cm. They are still in the ratio 1 : 2.

Key words

enlargement
a transformation that multiplies every distance from a centre by the same number. The shape keeps its form.
centre of enlargement
the point the distances are measured from. It does not move.
scale factor
the number every length is multiplied by. A scale factor between 0 and 1 makes the shape smaller.

Back to your guess

The answer is A: 6 squares right of the point. Every distance from the point doubles.

An enlargement multiplies every distance from the centre by the scale factor. A distance of 3 squares becomes 2 × 3 = 6 squares. Adding 2 would not keep the shape the same.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Triangle A is enlarged by scale factor 3 about C at (1, 1). Where does the corner (3, 2) go?

    A grid with an x axis and a y axis from 0 to 8. The centre C at (1, 1) and triangle A with corners at (3, 2), (4, 2) and (3, 3), labelled A.12345678123456780xyCA
    Tap your answer.

    1. Not this one. You added 3 to each number. An enlargement multiplies the distances from the centre by the scale factor. Go to M4.4.3. Try another.

    2. Yes, that is right. The corner (3, 2) is 2 right and 1 up from C. Multiply by 3: 6 right and 3 up. Counting from C at (1, 1) gives (7, 4).

    3. Not this one. You multiplied only the right step. Both numbers of the step from the centre are multiplied. Go to M4.4.3. Try another.

    See the answer

    (7, 4)

    The corner (3, 2) is 2 right and 1 up from C. Multiply by 3: 6 right and 3 up. Counting from C at (1, 1) gives (7, 4).

    • If you chose A: You added 3 to each number. An enlargement multiplies the distances from the centre by the scale factor. Go to M4.4.3.
    • If you chose C: You multiplied only the right step. Both numbers of the step from the centre are multiplied. Go to M4.4.3.

  2. The corner (6, 4) is enlarged by scale factor 1/2 about C at (2, 2). Where does it go?

    A grid with an x axis and a y axis from 0 to 8. The centre C at (2, 2) and triangle A with corners at (4, 2), (6, 2) and (6, 4), labelled A.12345678123456780xyCA
    Tap your answer.

    1. Not this one. You halved the numbers of the corner itself. Halve the step from the centre instead, then count from the centre. Go to M4.4.3. Try another.

    2. Not this one. You doubled the step from the centre. A scale factor of 1/2 halves it. Go to M4.4.3. Try another.

    3. Yes, that is right. The corner (6, 4) is 4 right and 2 up from C. Half of that is 2 right and 1 up. Counting from C at (2, 2) gives (4, 3).

    See the answer

    (4, 3)

    The corner (6, 4) is 4 right and 2 up from C. Half of that is 2 right and 1 up. Counting from C at (2, 2) gives (4, 3).

    • If you chose A: You halved the numbers of the corner itself. Halve the step from the centre instead, then count from the centre. Go to M4.4.3.
    • If you chose B: You doubled the step from the centre. A scale factor of 1/2 halves it. Go to M4.4.3.

  3. A triangle has a side of 4 cm. It is enlarged by scale factor 3. Which statement is true?

    Tap your answer.

    1. Not this one. You added 3 cm. An enlargement multiplies each length by the scale factor. Go to M4.4.3. Try another.

    2. Not this one. The side is right, but angles do not change in an enlargement. The shape keeps its form. Go to M4.4.3. Try another.

    3. Yes, that is right. Lengths are multiplied by the scale factor: 4 × 3 = 12 cm. Angles do not change.

    See the answer

    The side is 12 cm and the angles stay the same.

    Lengths are multiplied by the scale factor: 4 × 3 = 12 cm. Angles do not change.

    • If you chose A: You added 3 cm. An enlargement multiplies each length by the scale factor. Go to M4.4.3.
    • If you chose B: The side is right, but angles do not change in an enlargement. The shape keeps its form. Go to M4.4.3.

  4. Triangle A is enlarged to triangle A′. Where is the centre of the enlargement?

    A grid with an x axis and a y axis from 0 to 8. Triangle A with corners at (3, 2), (4, 2) and (3, 3), labelled A, and triangle A′ with corners at (4, 3), (6, 3) and (4, 5), labelled A′.12345678123456780xyAA′
    Tap your answer.

    1. Not this one. This is a corner of A. The centre is where the lines through matching corners meet, and it is not a corner of A. Go to M4.4.3. Try another.

    2. Not this one. This is a corner of A′. The centre is where the lines through matching corners meet. Go to M4.4.3. Try another.

    3. Not this one. Draw the line through (3, 2) and (4, 3) and the line through (4, 2) and (6, 3). They do not cross at this point. Go to M4.4.3. Try another.

    4. Yes, that is right. The line through (3, 2) and (4, 3) and the line through (4, 2) and (6, 3) cross at (2, 1). That is the centre.

    See the answer

    (2, 1)

    The line through (3, 2) and (4, 3) and the line through (4, 2) and (6, 3) cross at (2, 1). That is the centre.

    • If you chose A: This is a corner of A. The centre is where the lines through matching corners meet, and it is not a corner of A. Go to M4.4.3.
    • If you chose B: This is a corner of A′. The centre is where the lines through matching corners meet. Go to M4.4.3.
    • If you chose C: Draw the line through (3, 2) and (4, 3) and the line through (4, 2) and (6, 3). They do not cross at this point. Go to M4.4.3.

  5. A side of 10 cm is enlarged by scale factor 1/2. How long is the image side?

    Tap your answer.

    1. Yes, that is right. Multiply the length by the scale factor: 10 × 1/2 = 5 cm.

    2. Not this one. You doubled the length. A scale factor of 1/2 is half, so the image is smaller than the original. Go to M4.4.3. Try another.

    3. Not this one. You took 1/2 away. Scale factors multiply. Half of 10 cm is 5 cm. Go to M4.4.3. Try another.

    See the answer

    5 cm

    Multiply the length by the scale factor: 10 × 1/2 = 5 cm.

    • If you chose B: You doubled the length. A scale factor of 1/2 is half, so the image is smaller than the original. Go to M4.4.3.
    • If you chose C: You took 1/2 away. Scale factors multiply. Half of 10 cm is 5 cm. Go to M4.4.3.

  6. From an earlier chapter

    The point (1, 3) is translated by the vector with 5 on top and 2 below. What is the y number of the image?

    See the answer

    5

    Add the bottom number to y: 3 + 2 = 5.

  7. The corner (2, 5) is enlarged by scale factor 3 about O at (0, 0). What is the y number of the image?

    See the answer

    15

    The corner is 2 right and 5 up from O. Multiply by 3: 6 right and 15 up. The y number is 15.

  8. The corner (5, 3) is enlarged by scale factor 2 about C at (1, 1). What is the x number of the image?

    See the answer

    9

    The corner is 4 right of C. Double it: 8 right of C. Counting from x = 1 gives 1 + 8 = 9.

  9. A side of 5 cm becomes 15 cm in an enlargement. What is the scale factor?

    See the answer

    3

    The scale factor is the new length divided by the old length: 15 ÷ 5 = 3.

Keep it

Copy this into your exercise book.

Enlarge by multiplying each distance from the centre by the scale factor. Image = centre + factor × (corner − centre). Lines through matching corners meet at the centre.

Draw: A small triangle, a bigger one, and dashed lines from the centre through matching corners.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.0 MB). It opens on any phone.