KS3 · Maths

Rotating shapes

A roundabout, a door handle, the hands of a clock: each turns about one fixed point that never moves. Find that point and every rotation makes sense.

Year 7 · Geometry

Pin the centre, then turn
2.2 cm0°5.1 cm0°3.2 cm3.2 cm98°98°OABCA′B′C′PQ

O to P 2.2 cm. P has turned 0°. O to Q 5.1 cm. Q has turned 0°. AB 3.2 cm. A′B′ 3.2 cm. Angle B 98°. Angle B′ 98°. Classification: Rotation · 90° clockwise about O. Relationship: P lands on A′ at 90° and Q lands on C′ at 90°: every point turns through the same angle. Q swings along a much longer path because it is further from O, but it doesn't turn any further. O to P and O to Q never change as you drag — each point stays the same distance from the centre. And AB = A′B′ and angle B = angle B′: the image is congruent to the object.

O to P2.2 cmP has turned0°O to Q5.1 cmQ has turned0°AB3.2 cmA′B′3.2 cmAngle B98°Angle B′98°Drag P onto A′, then Q onto C′

Rotation · 90° clockwise about O

P lands on A′ at 90° and Q lands on C′ at 90°: every point turns through the same angle. Q swings along a much longer path because it is further from O, but it doesn't turn any further. O to P and O to Q never change as you drag — each point stays the same distance from the centre. And AB = A′B′ and angle B = angle B′: the image is congruent to the object.

Triangle ABC has been turned a quarter turn (90°) clockwise about the centre O, making the image A′B′C′. P starts on A and Q starts on C. Drag each one clockwise round O until it lands on its image point, and watch the readouts as you go.

Spot the rotation

Turned, flipped, or changed?

Is each image a rotation of its object? Place every pair, then read why.

Still to sort

A rotation (0)

Same lengths and angles, and it reads normally when you tilt your head

Where the line is: A mirror image keeps every length and angle too. The difference is that a mirror image is back to front, and no amount of turning fixes that.

A mirror image, so not a rotation (0)

Same lengths and angles, but back to front however you tilt it

Where the line is: If turning your head makes it read normally, it was turned, not flipped.

Shape or size changed, so not a rotation (0)

At least one length or angle is different

7 of 7 still to sort.

How big is the turn?

Place each turn on the 0°–360° line

Drag each turn to where you think it sits. Use the landmarks: a quarter turn is 90°, a half turn 180°, a three-quarter turn 270° and a full turn 360°.

Predict, then check

Clockwise is the way clock hands move: an arrow pointing up, turned clockwise, points right after a quarter turn, then down, then left. Anti-clockwise is the opposite way.

A flag is turned 90° clockwise about a point. Which anti-clockwise turn about the same point puts it in exactly the same place?

Rotating with tracing paper

Put the method in order

The order you do each step in when you rotate a shape with tracing paper

1 · First step6 · Last step

  1. Press firmly on each vertex to leave a dent in the page

  2. Lay tracing paper over the shape and the centre, and trace both

  3. Take the tracing paper away and join the dents to draw the image

  4. Push your pencil point firmly into the centre so it acts as a pivot

  5. Plot the centre of rotation, if it's given as a coordinate

  6. Turn the tracing paper through the given angle, in the given direction

Describe a rotation fully

Problem

Triangle PQR has vertices P(1, 1), Q(3, 1) and R(1, 2). Its image has vertices P′(5, 5), Q′(3, 5) and R′(5, 4). Describe the transformation fully.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Pin one point and turn everything else around it. The shape keeps every length and angle — only its position and the way it faces change.

What you need to know

  • A rotation turns an object by a fixed amount about a fixed point, the centre of rotation.
  • Lengths and angles are invariant: the image is congruent to the object, and it is never a mirror image.
  • A quarter turn is 90°, a half turn 180°, a three-quarter turn 270° and a full turn 360°.
  • A full description gives the direction, the size in degrees and the centre as a coordinate.

The big picture

Rotating shapes is all about one fixed point. A rotation — a turn, a spin — moves a shape by a fixed amount about that point, the centre of rotation. Every length and angle stays the same, so the image is congruent to the object, and a rotation is never a mirror image. Turns are measured in degrees, clockwise or anti-clockwise, and the centre decides where the image lands. To describe a rotation fully, give the direction, the size and the centre.

Key points

1The centre of rotation is the fixed point everything turns about. It can be on a vertex, inside the shape or outside it, and it fixes where the image lands.
2Every point stays the same distance from the centre as it turns. A vertex on the centre does not move at all.
3For the same angle, a point further from the centre travels further — it does not turn through a bigger angle.
4Invariant means unchanged by the transformation. Under rotation, lengths, angles and the order of the angles are invariant, so the image is congruent. Check with a ruler and a protractor.
5A rotated word can still be read by tilting your head; a mirrored word stays back to front. Unless the turn is a full turn (or several), the image faces a different way.
6Clockwise is the way clock hands move. The same rotation the other way round is the rest of the full turn: 100° clockwise = 360° − 100° = 260° anti-clockwise. A half turn is the same either way.
7Rotations can be more than a full turn: one and a quarter turns is 450°, which lands exactly where 90° does. Adding full turns never changes the image.
8Repeat the same rotation and you get more congruent images: four rotations of 90° make a full turn, so the fourth image lies back on the object.
9To find the centre of a 180° rotation, join at least two pairs of corresponding points; the centre is where the segments cross. This only works for 180°.

Worked example

Problem

Triangle ABC has vertices A(4, 1), B(4, 2) and C(2, 1). Rotate it 90° anti-clockwise about the centre (2, 1). Give the coordinates of the image.

⚠ Watch out

Giving only an angle, like '90° anti-clockwise'. The same turn about a different centre lands the image somewhere else, so always give the centre. And a far-away centre doesn't make a bigger turn — the shape just travels further through the same angle.

🧠

Memory hook

Tilt your head: a rotated word still reads, a mirrored word never does. And to go the other way round, take the rest of the full turn — 90° clockwise is 270° anti-clockwise.

✓

Check yourself

Could you describe a rotation fully without looking back — all four parts? And can you say which one turn gives the same image whichever direction you go?

Flashcards

(14)
What is a rotation?
Turning an object by a fixed amount around a fixed point, called the centre of rotation.
What does "invariant" mean?
A property that stays the same after a transformation. Under rotation, lengths and angles are invariant, so the image is congruent: same shape, same size.
What happens if you rotate a shape 90° four times about the same centre?
Each image is congruent to the object, and four quarter turns make a full turn, so the last image lies back on the object.
How can you tell a rotation from a mirror image?
Tilt your head: a word on a rotated shape still reads normally. A mirrored word stays back to front.
When do the object and image face the same way?
Only after a full turn of 360°, or a whole number of full turns. Even a square's corners move otherwise.
Which way is clockwise?
The way the hands of a clock move: the top of the object moves right and the bottom moves left.
Quarter, half, three-quarter and full turn in degrees?
90°, 180°, 270° and 360°.
Which rotation gives the same image clockwise or anti-clockwise?
180°, a half turn.
How do you describe a rotation the other way round?
Use the rest of the full turn: subtract from 360°. For example, 100° clockwise = 260° anti-clockwise.
How many degrees is two and three-quarter turns?
360° + 360° + 270° = 990°. It gives the same image as a three-quarter turn.
What four things fully describe a rotation?
That it's a rotation, the direction, the size in degrees, and the centre as a coordinate.
How do you find the centre of a 180° rotation?
Join at least two pairs of corresponding points. The centre is where the segments cross. This only works for 180°.
What does the centre of rotation decide?
Where the image lands. Every point stays the same distance from it, and a vertex on the centre does not move.
Why keep the tracing paper pinned while you turn it?
The pencil on the centre is the pivot. Lift the paper off and the image loses its correct position.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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