Rotations about a point | Year 7

01 / LEARN

Understand the idea.

A rotation needs three instructions

A rotation turns a shape about a fixed centre. Describe the centre, angle and direction, clockwise or anticlockwise. For a half-turn, either direction gives the same image. The centre itself does not move.

Each point travels along part of a circle centred at the rotation centre. Its distance from the centre stays unchanged. Check both the turning direction and the distance; matching only one of them does not locate the image.

Use the grid as a turning guide

About the origin, a $90^{\circ}$ anticlockwise turn sends $(x, y)$ to $(- y, x)$ ; a clockwise turn sends it to $(y, - x)$ . A $180^{\circ}$ turn reverses both signs. A sketch helps you check which quadrant the image should occupy.

For a different centre, first subtract the centre coordinates to describe the point relative to it. Rotate that displacement, then add the centre coordinates back. The familiar origin rules turn displacements, so they cannot be applied directly to coordinates about another centre.

The sage triangle has A(1,1), B(3,1), C(1,2). The dashed gold image has A prime (-1,1), B prime (-1,3), C prime (-2,1), after a ninety-degree anticlockwise rotation about the origin. Both coordinate directions use the same physical unit scale.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Quarter-turn about the origin

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

Worked solution

1
Select the anticlockwise quarter-turn rule.
$(x, y) \mapsto (- y, x)$
The stated centre is the origin, so the coordinate pair is already a displacement from it.
2
Substitute both coordinates.
$(- y, x) = (- (- 2), 3) = (2, 3)$
Reversing negative two makes the new horizontal coordinate positive.
3
Check the direction and label the image.
$P^{'} (2, 3)$
Three right and two down turns to two right and three up.

Answer

$P^{'} (2, 3)$

EXAMPLE 02

Use the stated centre

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

Equal-scale coordinate grid showing centre C(1,1), original point P(3,1) in sage, and image P prime (1,3) in gold after a ninety-degree anticlockwise rotation about C.

Worked solution

1
Measure P relative to the centre C.
$(3 - 1, 1 - 1) = (2, 0)$
The point starts two units right of C, rather than three units right of the origin.
2
Turn that displacement anticlockwise.
$(2, 0) \mapsto (0, 2)$
A quarter-turn changes a rightward movement into an upward movement.
3
Place the turned displacement back at C.
$P^{'} = (1 + 0, 1 + 2) = (1, 3)$
The image remains two units from the same rotation centre.

Answer

$P^{'} (1, 3)$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$(2, - 3)$

A half-turn about the origin reverses both displacements: $(- 2, 3) \mapsto (2, - 3)$ . The image stays the same distance from the origin.

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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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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 ↗

In this lesson

Understand the idea.

A rotation needs three instructions

Use the grid as a turning guide

See the method in action.

Quarter-turn about the origin

Select the anticlockwise quarter-turn rule.

Substitute both coordinates.

Check the direction and label the image.

Use the stated centre

Measure P relative to the centre C.

Turn that displacement anticlockwise.

Place the turned displacement back at C.

Try it yourself.

Check your understanding

Keep practising