Chapter 1 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

M4.4.1 Chapter 1

Reflections and rotations

I flip a shape over the line y = x. Where does the corner at (1, 3) 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

  • Reflect a shape in a line like y = x
  • Rotate a shape by 90° or 180° about a point
  • Describe a rotation by its centre, angle and direction

A shape on a grid can be flipped, turned, slid or made bigger. Each of these is a transformation. This chapter has the flip and the turn. The next two chapters have the slide and the enlargement.

A reflection flips a shape over a mirror line. The new shape is called the image. Each point and its image are the same distance from the mirror line, on opposite sides. The line joining them meets the mirror line at a right angle.

Draw it first

Reflect triangle A in the line x = 1

  1. A grid with an x axis and a y axis from −5 to 5. Step 1: triangle A with corners at (3, 1), (5, 2) and (3, 4), labelled A, and the dashed mirror line x = 1. Step 2: dashed lines from each corner straight across the mirror line. Step 3: the image, triangle A′, with corners at (−1, 1), (−3, 2) and (−1, 4), labelled A′.−4−224−4−2240xyx = 1A

    1 Triangle A has corners at (3, 1), (5, 2) and (3, 4). Draw the mirror line x = 1. Every point on it has x equal to 1.

  2. A grid with an x axis and a y axis from −5 to 5. Step 1: triangle A with corners at (3, 1), (5, 2) and (3, 4), labelled A, and the dashed mirror line x = 1. Step 2: dashed lines from each corner straight across the mirror line. Step 3: the image, triangle A′, with corners at (−1, 1), (−3, 2) and (−1, 4), labelled A′.−4−224−4−2240xyx = 1A

    2 From each corner, draw a line straight across the mirror. Count how far the corner is from the mirror.

  3. A grid with an x axis and a y axis from −5 to 5. Step 1: triangle A with corners at (3, 1), (5, 2) and (3, 4), labelled A, and the dashed mirror line x = 1. Step 2: dashed lines from each corner straight across the mirror line. Step 3: the image, triangle A′, with corners at (−1, 1), (−3, 2) and (−1, 4), labelled A′.−4−224−4−2240xyx = 1AA′

    3 Go the same distance past the mirror. Join the new points to make the image, A′.

The corner (3, 1) is 2 units from the line x = 1. So its image is 2 units on the other side, at (−1, 1). A point that lies on the mirror line is its own image.

Now take a mirror line that slants. The line y = x passes through (1, 1), (2, 2) and (3, 3). Every point on it has equal x and y.

A grid with an x axis and a y axis from 0 to 6. The dashed mirror line y = x slants up to the right. Triangle PQR has corners P at (1, 3), Q at (1, 5) and R at (2, 3), above the line. Its image, triangle P′Q′R′, is below the line, with P′ at (3, 1), Q′ at (5, 1) and R′ at (3, 2). The labels are P, Q, R, P′, Q′ and R′.1234561234560xyy = xPQRP′Q′R′
The mirror line y = x. Triangle PQR and its image lie on opposite sides.

The line y = x makes the same angle with both axes. Folding along it swaps the two axes, so a distance across becomes a distance up. A point (a, b) goes to the point (b, a). The numbers swap.

A grid with an x axis and a y axis from 0 to 6. The dashed mirror line y = x. Triangle A has corners (2, 4), (2, 6) and (3, 6), labelled A. Its image has corners (4, 2), (6, 2) and (6, 3), labelled A′.1234561234560xyy = xAA′
The triangle with corners (2, 4), (2, 6) and (3, 6), and its image in y = x.

Worked example

Reflect the triangle with corners (2, 4), (2, 6) and (3, 6) in the line y = x.

  1. Swap the two numbers of each corner.
  2. The corner (2, 4) goes to (4, 2).
  3. The corner (2, 6) goes to (6, 2).
  4. The corner (3, 6) goes to (6, 3).

Answer: (4, 2), (6, 2) and (6, 3)

A rotation turns a shape about one fixed point. That point is the centre of rotation. A turn can be clockwise, the way the hands of a clock move, or anticlockwise, the other way. A half turn of 180° needs no direction, because both ways end in the same place.

Draw it first

Turn triangle A by 90° clockwise about O

  1. A grid with an x axis and a y axis from −5 to 5. Step 1: triangle A with corners at (1, 1), (4, 1) and (1, 3), labelled A, and the centre O at (0, 0), a dashed line from O to the corner (4, 1), an arc with an arrowhead running clockwise down to the point (1, −4), and the label 90°. Step 2: the image, triangle A′, with corners at (1, −1), (1, −4) and (3, −1), labelled A′.−4−224−4−2240xyO90°A

    1 Triangle A has corners at (1, 1), (4, 1) and (1, 3). The centre of rotation is O at (0, 0). Take the corner (4, 1). Join it to O, then turn that line 90° clockwise. The corner lands at (1, −4).

  2. A grid with an x axis and a y axis from −5 to 5. Step 1: triangle A with corners at (1, 1), (4, 1) and (1, 3), labelled A, and the centre O at (0, 0), a dashed line from O to the corner (4, 1), an arc with an arrowhead running clockwise down to the point (1, −4), and the label 90°. Step 2: the image, triangle A′, with corners at (1, −1), (1, −4) and (3, −1), labelled A′.−4−224−4−2240xyO90°AA′

    2 Turn every corner the same way. Join the new corners to make the image, A′.

A turn keeps every point the same distance from the centre. The shape does not change size and does not flip over.

A grid with an x axis and a y axis from −5 to 5. The point P at (3, 1) and the point P′ at (1, −3). A dashed line from O at (0, 0) to each point, and an arc with an arrowhead running clockwise from P to P′. The labels are P, P′, O and 90°.−4−224−4−2240xyPP′O90°
P is 3 right and 1 up from O. After a 90° clockwise turn, P′ is 1 right and 3 down.

Look at P. Going 3 right and 1 up from O becomes going 1 right and 3 down. The right step turned into a down step, and the up step turned into a right step. So a 90° clockwise turn about O sends (x, y) to (y, −x).

A 90° anticlockwise turn about O sends (x, y) to (−y, x). A half turn of 180° about O sends (x, y) to (−x, −y). Each is the same step, turned by the angle.

A grid with an x axis and a y axis from −5 to 5. The centre C at (2, 1). Triangle A with corners at (3, 2), (5, 2) and (5, 5), labelled A. A dashed line from C to the corner (5, 2), an arc with an arrowhead running anticlockwise up to (1, 4), and the label 90°. The image, triangle A′, has corners at (1, 2), (1, 4) and (−2, 4), labelled A′.−4−224−4−2240xyC90°AA′
Triangle A turned 90° anticlockwise about C at (2, 1).

Worked example

Turn the triangle with corners (3, 2), (5, 2) and (5, 5) by 90° anticlockwise about C at (2, 1).

  1. Find how far each corner is from C. The corner (3, 2) is 1 right and 1 up. The corner (5, 2) is 3 right and 1 up. The corner (5, 5) is 3 right and 4 up.
  2. A 90° anticlockwise turn changes "right r, up u" into "left u, up r". So (3, 2) is now 1 left and 1 up from C.
  3. The corner (3, 2) goes to (1, 2). The corner (5, 2) goes to (1, 4). The corner (5, 5) goes to (−2, 4).

Answer: (1, 2), (1, 4) and (−2, 4)

To describe a rotation, say all three things: the centre, the angle and the direction. For a half turn, the direction can be left out. "90° clockwise about (0, 0)" is complete. "90° clockwise" is not.

A grid with an x axis and a y axis from −5 to 5. Triangle A with corners at (1, 1), (4, 1) and (1, 3), labelled A. A dashed straight line from the corner (4, 1) through the centre O at (0, 0) to the corner (−4, −1), with an arc running round and the label 180°. The image, triangle A′, has corners at (−1, −1), (−4, −1) and (−1, −3), labelled A′.−4−224−4−2240xyO180°AA′
A half turn about O. The same image comes from turning clockwise or anticlockwise.

See it three ways

With things

Fold a square sheet along its diagonal. Hold it to the light. A dot on one half shows through on the other half, exactly where its reflection is.

As a picture

A grid with an x axis and a y axis from 0 to 6. The dashed mirror line y = x. The point P at (1, 3) above the line and the point P′ at (3, 1) below it, joined by a dashed line that crosses the mirror line at a right angle.1234561234560xyy = xPP′

In symbols

A reflection in y = x sends (a, b) to (b, a). A 90° clockwise turn about O sends (x, y) to (y, −x).

Try it

Fold along y = x

You need: a square sheet of thin paper, a pencil, a ruler.

  1. Draw an x axis and a y axis along two edges of the paper, with the origin in one corner. Mark every 1 cm.
  2. Draw the line from the origin to the opposite corner. That is the line y = x.
  3. Mark the point (1, 3) with a dot. Fold the paper along y = x, so the two halves lie on top of each other.
  4. Hold the folded sheet up to the light. Put a second dot on the top half, exactly over the dot that shows through.

What you will see: Open the sheet. The second dot is at (3, 1). The fold swapped the two numbers.

If you cannot try it, this is what happens.
A table with the head Dot, Before and After, for the fold. Row 1: P, (1, 3) and (3, 1).DotBeforeAfterP(1, 3)(3, 1)

A wrong idea many people have

A 90° turn about (0, 0) gives the same picture as a reflection in y = x.

Why it feels right: Both moves keep the size of the shape, and both put the numbers of a point in a new order.

The test that breaks it: Take the corner (1, 3). Turn it 90° clockwise about (0, 0) and it goes to (3, −1). Reflect it in y = x and it goes to (3, 1). These are different points.

What is true: A reflection flips the shape over, so its corners run round the other way. A rotation does not flip it. They are different transformations.

Key words

transformation
a rule that moves a shape: a flip, a turn, a slide or an enlargement.
reflection
a flip of a shape over a mirror line.
mirror line
the line a shape is flipped over. A point and its image are the same distance from it.
image
the shape that a transformation makes.
rotation
a turn of a shape about a fixed point.
centre of rotation
the point that stays still while the shape turns.
clockwise
in the direction the hands of a clock move.
anticlockwise
in the opposite direction to the hands of a clock.

Back to your guess

The answer is B: To (3, 1). The two numbers swap.

The line y = x runs through every point with equal x and y, such as (2, 2). A flip over it swaps the two numbers, so (1, 3) goes to (3, 1). The other two answers are flips over the x axis and over the y axis.

Check yourself

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

  1. Triangle A has a corner at (2, 5). A is reflected in the line y = x. Where is that corner now?

    Tap your answer.

    1. Not this one. You changed the sign of x. That is a reflection in the y axis. A reflection in y = x swaps the two numbers. Go to M4.4.1. Try another.

    2. Yes, that is right. The mirror line y = x swaps the two numbers of every point. The corner (2, 5) goes to (5, 2).

    3. Not this one. You changed the sign of y. That is a reflection in the x axis. A reflection in y = x swaps the two numbers. Go to M3.9.3. Try another.

    See the answer

    (5, 2)

    The mirror line y = x swaps the two numbers of every point. The corner (2, 5) goes to (5, 2).

    • If you chose A: You changed the sign of x. That is a reflection in the y axis. A reflection in y = x swaps the two numbers. Go to M4.4.1.
    • If you chose C: You changed the sign of y. That is a reflection in the x axis. A reflection in y = x swaps the two numbers. Go to M3.9.3.

  2. Triangle A is reflected in the mirror line x = 3. Its corner at (1, 2) is marked. Where is the image of that corner?

    A grid with an x axis and a y axis from −5 to 5. The dashed mirror line x = 3. Triangle A with corners at (1, 2), (2, 1) and (2, 4), labelled A, on the left of the mirror line. A dot marks the corner at (1, 2).−4−224−4−2240xyx = 3A
    Tap your answer.

    1. Not this one. That is the reflection in the y axis, the line x = 0. The mirror here is the line x = 3. Go to M4.4.1. Try another.

    2. Not this one. The image is not on the mirror line. It is as far past the line as the corner is before it, which is 2 units. Go to M4.4.1. Try another.

    3. Yes, that is right. The corner (1, 2) is 2 units left of the line x = 3. Its image is 2 units right of the line, at (5, 2). The y number stays the same.

    See the answer

    (5, 2)

    The corner (1, 2) is 2 units left of the line x = 3. Its image is 2 units right of the line, at (5, 2). The y number stays the same.

    • If you chose A: That is the reflection in the y axis, the line x = 0. The mirror here is the line x = 3. Go to M4.4.1.
    • If you chose B: The image is not on the mirror line. It is as far past the line as the corner is before it, which is 2 units. Go to M4.4.1.

  3. Triangle A is turned 90° clockwise about O at (0, 0). Where does the corner (4, 1) go?

    A grid with an x axis and a y axis from −5 to 5. Triangle A with corners at (1, 1), (4, 1) and (1, 3), labelled A, and the point O at (0, 0).−4−224−4−2240xyOA
    Tap your answer.

    1. Yes, that is right. A 90° clockwise turn about O sends (x, y) to (y, −x). So (4, 1) goes to (1, −4).

    2. Not this one. You turned it the other way, anticlockwise. Clockwise is the way the hands of a clock move. Go to M4.4.1. Try another.

    3. Not this one. That is a half turn, 180°. A quarter turn of 90° is half as much. Go to M4.4.1. Try another.

    See the answer

    (1, −4)

    A 90° clockwise turn about O sends (x, y) to (y, −x). So (4, 1) goes to (1, −4).

    • If you chose B: You turned it the other way, anticlockwise. Clockwise is the way the hands of a clock move. Go to M4.4.1.
    • If you chose C: That is a half turn, 180°. A quarter turn of 90° is half as much. Go to M4.4.1.

  4. Triangle A is moved to triangle B by a rotation. Which rotation is it?

    A grid with an x axis and a y axis from −5 to 5. Triangle A with corners at (1, 1), (3, 1) and (1, 4), labelled A, and triangle B with corners at (−1, 1), (−1, 3) and (−4, 1), labelled B. The point O at (0, 0).−4−224−4−2240xyOAB
    Tap your answer.

    1. Not this one. Clockwise would put the image below the x axis. B is above it, so the turn is the other way. Go to M4.4.1. Try another.

    2. Not this one. A half turn would put B below and to the left of O, in the opposite corner of the grid. Go to M4.4.1. Try another.

    3. Yes, that is right. A turn of 90° anticlockwise about O sends (x, y) to (−y, x). The corner (1, 1) goes to (−1, 1) and (3, 1) goes to (−1, 3). Those are corners of B.

    4. Not this one. The angle and the direction are right, but the centre is not. Every point of A is turned about O, and O stays still. Go to M4.4.1. Try another.

    See the answer

    90° anticlockwise about O

    A turn of 90° anticlockwise about O sends (x, y) to (−y, x). The corner (1, 1) goes to (−1, 1) and (3, 1) goes to (−1, 3). Those are corners of B.

    • If you chose A: Clockwise would put the image below the x axis. B is above it, so the turn is the other way. Go to M4.4.1.
    • If you chose B: A half turn would put B below and to the left of O, in the opposite corner of the grid. Go to M4.4.1.
    • If you chose D: The angle and the direction are right, but the centre is not. Every point of A is turned about O, and O stays still. Go to M4.4.1.

  5. Which description of a rotation is complete?

    Tap your answer.

    1. Not this one. This gives the angle and direction but not the centre. Different centres give different images. Go to M4.4.1. Try another.

    2. Not this one. This gives the centre but not the angle or the direction. Go to M4.4.1. Try another.

    3. Not this one. This gives the angle and the centre but not the direction. A 90° turn can go either way. Go to M4.4.1. Try another.

    4. Yes, that is right. A rotation needs a centre, an angle and a direction. Only the last choice gives all three.

    See the answer

    90° clockwise about (2, 1)

    A rotation needs a centre, an angle and a direction. Only the last choice gives all three.

    • If you chose A: This gives the angle and direction but not the centre. Different centres give different images. Go to M4.4.1.
    • If you chose B: This gives the centre but not the angle or the direction. Go to M4.4.1.
    • If you chose C: This gives the angle and the centre but not the direction. A 90° turn can go either way. Go to M4.4.1.

  6. A point is reflected in the line y = x. It starts at (7, 2). What is the x number of the image?

    See the answer

    2

    A reflection in y = x swaps the two numbers. (7, 2) goes to (2, 7), so the x number is 2.

  7. A corner at (1, 4) is reflected in the line x = 3. What is the x number of the image?

    See the answer

    5

    The corner is 2 units left of the line x = 3. The image is 2 units right of it, so its x number is 3 + 2 = 5.

  8. A point is 4 units right of the centre C and 6 units above it. It makes a half turn about C. How many units below C is the image?

    See the answer

    6

    A half turn changes right into left and up into down. The image is 4 units left of C and 6 units below it.

  9. A point at (2, 6) is turned 90° clockwise about O at (0, 0). How many units below the x axis is the image?

    See the answer

    2

    A 90° clockwise turn about O sends (x, y) to (y, −x). So (2, 6) goes to (6, −2), which is 2 units below the x axis.

Keep it

Copy this into your exercise book.

A reflection flips a shape over a mirror line. In y = x the two numbers swap. A rotation needs a centre, an angle and a direction. A half turn needs no direction.

Draw: A triangle and its image in the line y = x, with the corners named.

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.