Rotations about a point | Year 7

01 / LEARN

The idea, step by step

A rotation needs three instructions

To describe a rotation, give the centre, the angle and the direction: clockwise or anticlockwise. For a half-turn, either direction gives the same image. The centre itself stays fixed.

Every point follows part of a circle around the centre. The original and image are the same distance from that centre. Do not rotate around the origin unless the origin is the stated centre.

Use the grid as a turning guide

For a quarter-turn about the origin, follow the horizontal and vertical steps from the origin to the point, then turn those steps. A 90° anticlockwise turn sends $(x, y)$ to $(- y, x)$ ; a 90° clockwise turn sends it to $(y, - x)$ . A 180° turn changes both signs.

For another centre, first describe the point relative to that centre. Turn that displacement, then put the turned steps back from the same centre. Check the direction with a sketch rather than swapping signs without thinking.

A common trap

Using the origin rule when the centre is not the origin.

Instead: Measure the displacement from the stated centre, turn it and return to that same centre.

02 / WORKED EXAMPLES

See the thinking

Example 1

Quarter-turn about the origin

Rotate $P (3, - 2)$ through 90° anticlockwise about the origin.

A point 3 right and 2 down turns to a point 2 right and 3 up.
Equivalently, use $(x, y) \mapsto (- y, x)$ .
$(- (- 2), 3) = (2, 3)$ .

P^{'} (2, 3)

Example 2

Use the stated centre

Rotate $P (3, 1)$ through 90° anticlockwise about $C (1, 1)$ .

From C to P is 2 units right and 0 units up.
Turn this displacement anticlockwise: 2 units right becomes 2 units up.
Starting at C, move 2 units up.

P^{'} (1, 3)

Check your understanding

Rotate $(- 2, 3)$ through 180° about the origin.

Show answer

$(2, - 3)$

A half-turn sends each coordinate to its opposite.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Rotate $(2, 4)$ 90° clockwise about the origin.
2 marks
Rotate $(- 3, 1)$ 90° anticlockwise about the origin.
2 marks
Rotate $(4, - 1)$ 180° about the origin.
2 marks
Rotate $P (4, 2)$ 90° anticlockwise about $C (2, 2)$ .
2 marks
Rotate $A (1, 1)$ , $B (3, 1)$ and $C (1, 2)$ 90° anticlockwise about the origin. Draw and label the image on the supplied grid.
3 marks
A point is the centre of a rotation. Where does that point go? Explain.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Rotate $(2, 4)$ 90° clockwise about the origin.
2 marks
A 90° clockwise turn uses $(x, y) \mapsto (y, - x)$ .
$(4, - 2)$
Rotate $(- 3, 1)$ 90° anticlockwise about the origin.
2 marks
A 90° anticlockwise turn uses $(x, y) \mapsto (- y, x)$ .
$(- 1, - 3)$
Rotate $(4, - 1)$ 180° about the origin.
2 marks
Change both signs for a half-turn.
$(- 4, 1)$
Rotate $P (4, 2)$ 90° anticlockwise about $C (2, 2)$ .
2 marks
Relative to C, P is 2 units right.
After the turn, it is 2 units above C.
$(2, 4)$
Rotate $A (1, 1)$ , $B (3, 1)$ and $C (1, 2)$ 90° anticlockwise about the origin. Draw and label the image on the supplied grid.
3 marks
Apply $(x, y) \mapsto (- y, x)$ to each vertex.
$A^{'} (- 1, 1), B^{'} (- 1, 3), C^{'} (- 2, 1)$
A point is the centre of a rotation. Where does that point go? Explain.
2 marks
Its distance from the centre is zero before and after every turn.
It stays at the same coordinates.

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Recording transformations with coordinates

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Translations; reflections in coordinate axes; rotations about a point; recording transformations with coordinates; sequences of transformations; explaining what stays unchanged.

Official curriculum source ↗

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