Year 9 · Unit 6 Transformations · Lesson 3 of 6

What three things describe a turn?

A rotation turns a shape about a centre through an angle in a direction. About the origin, quarter turns and half turns have simple coordinate rules, and moving the centre moves the image.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 2. Reflect the point (5, −2) in the y-axis.
    Answer(−5, −2). The y-axis changes the sign of x.
  2. From Year 7, Unit 4, lesson 2, angles on a line and at a point. How many degrees are in a full turn, and how many in a half turn?
    Answer360 degrees in a full turn and 180 degrees in a half turn.
  3. From Year 8, Unit 3, integers. What is the opposite of 4?
    Answer−4.
Words used in this lesson
rotation
a turn about a fixed point. For example: a wheel turning on its axle.
centre of rotation
the point that does not move when a shape turns. For example: the pin of a pinwheel.
clockwise
in the direction a clock hand moves. For example: from 12 towards 3.
anticlockwise
in the direction opposite to a clock hand. For example: from 12 towards 9.
quarter turn
a turn of 90 degrees. For example: from 12 to 3 on a clock.

A pinwheel at a market stall

At a market in Oaxaca, a paper pinwheel is pinned at the origin of a grid drawn on its box. One of its blades is the triangle ABC with corners A (−2, 1), B (1, 1) and C (−2, 3). The wind turns the blade a half turn, 180 degrees, about the pin.

Predict first: where does the corner C (−2, 3) land after a half turn about the origin? Then try the other two turns.

(−1, −2)image of A, A′
(−1, 1)image of B, B′
(−3, −2)image of C, C′

Turned 90° anticlockwise about the origin, A (−2, 1) lands on A′ (−1, −2).

Show me the steps
  1. 90° anticlockwise, rule (x, y) → (−y, x): A (−2, 1) goes to (−1, −2), B (1, 1) to (−1, 1) and C (−2, 3) to (−3, −2).
  2. 180°, rule (x, y) → (−x, −y): A goes to (2, −1), B to (−1, −1) and C to (2, −3). Both signs change.
  3. 90° clockwise, rule (x, y) → (y, −x): A goes to (1, 2), B to (1, −1) and C to (3, 2).
  4. The two quarter turns end up on opposite sides of the origin: from A, the clockwise image (1, 2) and the anticlockwise image (−1, −2) are a half turn apart.

A rotation needs three things: a centre (the one point that does not move), an angle and a direction, clockwise or anticlockwise.

About the origin: 90° anticlockwise is (x, y) → (−y, x); 180° is (x, y) → (−x, −y); 90° clockwise is (x, y) → (y, −x).

Every point stays the same distance from the centre as it turns, so lengths and angles are kept: a rotation is a congruence transformation.

The centre decides where it lands

The turn below is always 90° anticlockwise. Move the centre and watch the image.

1
1
(1, −2)image of A, A′
(1, 1)image of B, B′
(−1, −2)image of C, C′

Turned 90° anticlockwise about (1, 1), A (−2, 1) lands on A′ (1, −2). B is the centre, so it does not move.

Show me the steps
  1. Centre (1, 1): A goes to (1, −2), C to (−1, −2), and B (1, 1) is the centre, so it does not move.
  2. Centre (0, 0), the origin: A goes to (−1, −2), B to (−1, 1) and C to (−3, −2).
  3. Centre (−1, 1): A goes to (−1, 0), B to (−1, 3) and C to (−3, 0).
  4. The turn is the same each time, a quarter turn anticlockwise, and the image is the same shape. Only the centre changed, and the image moved with it.

Direction matters for a quarter turn. 90° clockwise and 90° anticlockwise send the same point to opposite places, 180° apart. A half turn has no direction to give: clockwise and anticlockwise are the same. To describe a rotation fully, give the centre, the angle and the direction.

Worked examples

Turn P (3, 2) through 90° anticlockwise about the origin. Use (x, y) → (−y, x): −2 across, 3 up. P′ (−2, 3). P is in the first quadrant and a quarter turn anticlockwise takes it to the second.

Turn the triangle (1, 1), (4, 1), (1, 3) through 180° about the origin. Change both signs: (−1, −1), (−4, −1), (−1, −3).

Turn Q (5, −2) through 90° clockwise about the origin. Use (x, y) → (y, −x): y is −2 and −x is −5. Q′ (−2, −5). Check: Q is in the fourth quadrant and a clockwise quarter turn takes it to the third.

Turn R (4, 3) through 90° anticlockwise about the centre (1, 1).

  1. Move the centre to the origin: R is 3 across and 2 up from the centre, so as a vector (3, 2).
  2. Turn the vector: (3, 2) → (−2, 3).
  3. Add the centre back: (1 + (−2), 1 + 3). R′ (−1, 4).

Your go. The first step is given. Turn S (−3, −1) through 180° about the origin.

  1. A half turn about the origin changes the sign of both coordinates.
  2. Write S′.
Check the answer

S′ is (3, 1).

Check before you go on

1. The point (4, −3) is turned 90° anticlockwise about the origin. Where does it land?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The point (6, 2) is turned 180° about the origin. Its image is (x, y). What is x?

3. The point (−5, 4) is turned 90° anticlockwise about the origin. What is the y-coordinate of the image?

Level 2 · Building

4. The point (3, −4) is turned 90° clockwise about the origin. What is the x-coordinate of the image?

5. A signpost corner at (7, 3) is turned 90° anticlockwise about the centre (2, 1). What is the x-coordinate of the image?

6. The point (4, 2) is moved to (−2, 4) by a rotation about the origin. Which rotation is it?

Level 3 · Stretch

7. The point (2, 5) is turned 90° anticlockwise about the origin, and the image is turned 90° anticlockwise again. The final image is (x, y). What is x + y?

8. A point is turned 90° anticlockwise about the origin and lands at (−6, 3). What was its x-coordinate before the turn?

Something to think about

A half turn (180 degrees) about the origin sends (x, y) to (−x, −y). Does it matter whether you turn it clockwise or anticlockwise? Why, or why not?

One way to think about it

No. Half a turn is 180 degrees whichever way you go, so a half turn clockwise and a half turn anticlockwise land on exactly the same point. Quarter turns are different: 90 degrees clockwise and 90 degrees anticlockwise end up on opposite sides of the origin.

Exit ticket

9. Which rule is a rotation of 180° about the origin?