Practice Maths

L63 — Rotations

Key Terms

rotation
A transformation that turns a shape around a fixed point by a given angle and direction. Size and shape are preserved.
centre of rotation
The fixed point around which the shape turns. For most questions this is the origin (0, 0).
clockwise
Turning in the same direction as clock hands — to the right at the top of the circle.
anticlockwise
Turning opposite to clock hands — to the left at the top. Also called counterclockwise.
↻
Clockwise
(same as clock hands)
↺
Anticlockwise
(opposite direction)
x-axis y-axis 1 2 3 −1 −2 −3 1 2 3 −1 −2 −3 O P(3, 2) (2, −3) 90° clockwise (−2, 3) 90° anticlockwise (−3, −2) 180°

All rotations of P(3, 2) about the origin O. Each point is the same distance from O.

Rotation Rules (Centre at the Origin)

  • 90° clockwise: (x, y) → (y, −x)
  • 90° anticlockwise: (x, y) → (−y, x)
  • 180° (either direction): (x, y) → (−x, −y)
  • 270° clockwise = 90° anticlockwise: (x, y) → (−y, x)
Hot Tip: 270° clockwise = 90° anticlockwise — always use the smaller equivalent angle. It saves time and reduces errors!

Worked Example 1 — 90° Clockwise

Rotate (3, 2) by 90° clockwise about the origin.

Rule: (x, y) → (y, −x). So (3, 2) → (2, −3).

Worked Example 2 — 90° Anticlockwise

Rotate (3, 2) by 90° anticlockwise about the origin.

Rule: (x, y) → (−y, x). So (3, 2) → (−2, 3).

Worked Example 3 — 180°

Rotate (3, 2) by 180° about the origin.

Rule: (x, y) → (−x, −y). So (3, 2) → (−3, −2).

x y (3,2) original (2,−3) 90° clockwise (−2,3) 90° anticlockwise (−3,−2) 180° O

All four positions of (3, 2) after different rotations about the origin.

What Is a Rotation?

A rotation turns a shape around a fixed point called the centre of rotation. Every point in the shape travels along a circular arc, staying the same distance from the centre. The shape doesn't change size or flip — it just turns.

Think of a wheel spinning around its hub, or clock hands rotating around the centre of the clock face. The shape (hand/spoke) turns but doesn't stretch or flip.

Describing a Rotation: Three Things Needed

To fully describe a rotation, you need to state three things:

  • Centre of rotation: the fixed point around which the shape turns (often the origin, a vertex, or another specified point)
  • Angle of rotation: how far it turns (e.g. 90°, 180°, 270°)
  • Direction: clockwise (like a clock hand moving forward) or anticlockwise (the opposite direction)

Example: "Rotate 90° clockwise about the origin."

Common Rotations About the Origin

When rotating a point (x, y) about the origin:

  • 90° clockwise: (x, y) → (y, −x)
  • 90° anticlockwise: (x, y) → (−y, x)
  • 180° (either direction): (x, y) → (−x, −y)

Example: Rotate (3, 2) by 90° clockwise: → (2, −3).

Locating the Centre of Rotation

Given a shape and its image after rotation, you can find the centre of rotation by connecting corresponding vertices of the pre-image and image with line segments, then finding the perpendicular bisector of each segment — they all meet at the centre of rotation. In practice at this level, the centre is usually given or visible on a grid.

Key tip: A 270° clockwise rotation gives the same result as a 90° anticlockwise rotation, and vice versa. So there are really only 3 distinct rotation amounts to know: 90°, 180°, and 270°. If in doubt, use tracing paper — trace the shape, place your pencil on the centre point, and turn the paper by the required angle.

  1. Rotate 90° Clockwise Fluency

    Rotate each point 90° clockwise about the origin. Rule: (x, y) → (y, −x). Blue = original; red = image; green arc = rotation path.

    4 4 1 −1 90° ↻ P(1, 4) P′(4, −1)
    (a) Rotate (1, 4)
    3 −2 −2 −3 90° ↻ P(3, −2) P′(−2, −3)
    (b) Rotate (3, −2)
    −5 1 1 5 90° ↻ P(−5, 1) P′(1, 5)
    (c) Rotate (−5, 1)
    6 6 90° ↻ P(0, 6) P′(6, 0)
    (d) Rotate (0, 6)

  2. Rotate 90° Anticlockwise Fluency

    Rotate each point 90° anticlockwise about the origin. Rule: (x, y) → (−y, x). Blue = original; red = image; green arc = rotation path.

    1 4 −4 1 90° ↺ P(1, 4) P′(−4, 1)
    (a) Rotate (1, 4)
    3 −2 2 3 90° ↺ P(3, −2) P′(2, 3)
    (b) Rotate (3, −2)
    −5 1 −1 −5 90° ↺ P(−5, 1) P′(−1, −5)
    (c) Rotate (−5, 1)
    6 −6 90° ↺ P(0, 6) P′(−6, 0)
    (d) Rotate (0, 6)

  3. Rotate 180° Fluency

    Rotate each point 180° about the origin. Rule: (x, y) → (−x, −y). Blue = original; red = image; green semicircle = rotation path.

    2 5 −2 −5 180° P(2, 5) P′(−2, −5)
    (a) Rotate (2, 5)
    −4 3 4 −3 180° P(−4, 3) P′(4, −3)
    (b) Rotate (−4, 3)
    1 −6 −1 6 180° P(1, −6) P′(−1, 6)
    (c) Rotate (1, −6)
    (0, 0) stays put! The origin maps to itself under any rotation about origin.
    (d) Rotate (0, 0)

  4. Rotate Shapes Fluency

    Find the image vertices after each rotation about the origin. Blue = original; red dashed = image.

    90° ↻ A B C A′ B′ C′
    (a) ABC 90° clockwise
    90° ↺ P Q R P′ Q′ R′
    (b) PQR 90° anticlockwise
    180° D E F G D′ E′ F′ G′
    (c) DEFG 180°
    270°↻ = 90°↺ J K L J′ K′ L′
    (d) JKL 270° clockwise

  5. Comparing Rotations Understanding

    1 1 −4 −4 1st 90°↻ 2nd 90°↻ (4, 1) (1, −4) (−4, −1)
    (a) Two 90° clockwise steps
    2 5 −5 2 270°↻ 90°↺ (2, 5) (−5, 2) same result!
    (b) 270°↻ vs 90°↺ from (2, 5)
    3 4 −3 −4 180° (3, 4) (−3, −4)
    (c) Map (3, 4) to (−3, −4)
    5 −2 2 5 90°↺ (5, −2) (2, 5)
    (d) Map (5, −2) to (2, 5)
    1. Point (4, 1) is rotated 90° clockwise, then 90° clockwise again. What is the final position? What single rotation gives the same result?
    2. Is 270° clockwise the same as 90° anticlockwise? Show using the point (2, 5).
    3. Describe the rotation that maps (3, 4) to (−3, −4).
    4. Describe the rotation that maps (5, −2) to (2, 5).

  6. Rotation Challenges Problem Solving

    90°↻ Start (0, 5) T1 (5, 0) T2 (0, −5) T3 (−5, 0) T4 returns to Start
    (a) Windmill blade tip
    1st↺ 2nd↺ A B C A′ B′ C′ A′′ B′′ C′′
    (b) Two 90° anticlockwise steps
    5 3 3 −5 90°↻ P(5, 3) P′(3, −5) P′ given; find P
    (c) Work backwards from image
    2 4 2 −4 1st: 90°↻ 2nd: 270°↻ (4, 2) returns! (2, −4)
    (d) A round-trip sequence
    1. A windmill blade has a tip at (0, 5). After each 90° clockwise turn, where is the tip? List the tip position for the first four turns.
    2. Triangle A(1, 0), B(3, 0), C(2, 2) is rotated 90° anticlockwise, and the image is then rotated 90° anticlockwise again. List the vertices after each rotation.
    3. A point P′(3, −5) is the image of point P after a 90° clockwise rotation about the origin. Find the original coordinates of P.
    4. Describe a sequence of two rotations about the origin that maps (4, 2) back to (4, 2).

  7. Mixed Rotation Drill Fluency

    For each point, apply all four rotations about the origin and record each image: 90° clockwise, 90° anticlockwise, 180°, and 270° clockwise.

    ● Blue = original  ● Orange = 90°↻  ● Green = 90°↺ & 270°↻  ● Red = 180°

    (1, 3) (3, −1) (−3, 1) (−1, −3) 270°↻ = 90°↺ same point
    (a) Rotate (1, 3)
    (4, −2) (−2, −4) (2, 4) (−4, 2) 270°↻ = 90°↺ same point
    (b) Rotate (4, −2)
    (−3, 5) (5, 3) (−5, −3) (3, −5) 270°↻ = 90°↺ same point
    (c) Rotate (−3, 5)
    (0, 7) (7, 0) (−7, 0) (0, −7) All 4 images lie on the axes!
    (d) Rotate (0, 7)
    1. (1, 3)
    2. (4, −2)
    3. (−3, 5)
    4. (0, 7)

  8. Identifying the Rotation Understanding

    In each case, describe the rotation (angle and direction) that maps the original point to the image. The centre of rotation is always the origin.

    2 3 −3 2 ? (2, 3) (−3, 2)
    (a) What rotation?
    5 1 −5 ? (5, 1) (−5, −1)
    (b) What rotation?
    4 −2 −2 −4 ? (4, −2) (−2, −4)
    (c) What rotation?
    3 4 4 −3 ? (3, 4) (4, −3)
    (d) What rotation?
    1. (2, 3) → (−3, 2)
    2. (5, 1) → (−5, −1)
    3. (4, −2) → (−2, −4)
    4. (3, 4) → (4, −3)

  9. Rotations — Properties and Patterns Understanding

    1 3 1 3 90°↻ A B C D A′ B′ C′ D′
    (a) Square 90° clockwise
    P (5, 0) T1 (0, −5) T2 (−5, 0) T3 (0, 5) T4 returns to P!
    (b) P(5, 0) cycled four times
    ① ② ③ ④ Start = End! 4 × 90° = 360° ✓ True
    (c) Four 90° turns = full circle
    180° (3, 4) A: (−3, −4) ✓ B: (4, 3) ✗ B confused 180° with y = x reflection
    (d) Which student is right?
    1. A square has vertices A(1,1), B(3,1), C(3,3), D(1,3). Rotate 90° clockwise about the origin. List the image vertices. Is the image the same shape and size as the original?
    2. Point P(5, 0) is rotated 90° clockwise, and then rotated 90° clockwise again three more times. List all four image positions. What do you notice?
    3. True or False: rotating a shape 90° clockwise four times brings it back to its starting position. Explain.
    4. Two students disagree: Student A says rotating (3, 4) by 180° gives (−3, −4). Student B says it gives (4, 3). Who is correct? Why?

  10. Rotations in Context Problem Solving

    12 3 6 9 15 s (0, 5) (5, 0)
    (a) Clock hand after 15 seconds
    (3, 1) (−1, 3) (−3, −1) (1, −3) 90°↺ 90°↺ 90°↺
    (b) Logo point rotated 3 times
    180° A B C A′ B′ C′ translate A′′ B′′ C′′
    (c) Rotate 180° then translate
    90°↻ 90°↻ 90°↻ N (0, 4) E (4, 0) S (0, −4) W (−4, 0)
    (d) Compass rose — each 90° clockwise
    1. A second-hand on a clock starts at the 12 o’clock position, which corresponds to the point (0, 5) on a coordinate grid. After 15 seconds (quarter turn clockwise), where is the tip of the hand?
    2. A logo is made by rotating an L-shaped piece 90° anticlockwise three times about the origin. If one point of the L is at (3, 1), find its position after each rotation (four positions total including the original).
    3. Triangle A(2, 1), B(4, 1), C(3, 3) is rotated 180° about the origin. The image is then translated right 3, up 2. Find the final vertices of the doubly transformed shape.
    4. A compass rose has north at (0, 4), east at (4, 0), south at (0, −4) and west at (−4, 0). Show that each direction is a 90° clockwise rotation of the previous one.
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