Linlithgow Academy · Mathematics · S2 Revision

Rotational Symmetry and Transformations

The order of rotational symmetry, completing shapes, reflections, translations and the equations of mirror lines.

Key ideas

Rotational symmetry

A shape has rotational symmetry if it looks exactly the same more than once as you turn it through one full turn.

The order is how many times it fits onto itself in one full turn.

A shape that only fits once has order 1 (no rotational symmetry).

Using tracing paper

Trace the shape, put your pencil on the centre and turn. Count how many times it matches.

Order 2: it matches every half turn. Order 4: every quarter turn.

Reflection

Each corner ends up the same distance from the mirror line, on the other side.

Reflect each corner, then join them up.

Translation

Every corner slides the same distance in the same direction: right or left, then up or down.

The shape does not turn or flip.

Equations of mirror lines

A vertical line through 3 on the x-axis is x = 3. A horizontal line through −1 on the y-axis is y = −1.

The y-axis is x = 0 and the x-axis is y = 0.

Order of rotational symmetry

Rotational Symmetry

Finding the order

What is the order of rotational symmetry of this design?(1 mark)
Count the matches in one full turn.It matches after ⅓ turn, ⅔ turn and a full turnOrder = 3✓1
In plain EnglishEach blade moves onto the next one. There are 3 identical blades, so it fits onto itself 3 times in a full turn.
Note: this design has no lines of symmetry, but it does have rotational symmetry. They are different things.
Marking points
M1 Order 3
Rotational Symmetry

A rectangle

What is the order of rotational symmetry of a rectangle?(1 mark)
Turn it.It matches after a half turn and after a full turnOrder = 2✓1
In plain EnglishAfter a quarter turn the rectangle is standing on its short side, so it doesn't match. It matches again after a half turn.
Marking points
M1 Order 2

Using rotational symmetry

Rotational Symmetry

Completing a shape

Shade more squares so the shape has rotational symmetry of order 2 about the red dot.(2 marks)
Turn a half turn.Each square moves to the opposite side of the dot, the same distance away
In plain EnglishImagine turning tracing paper a half turn (180°) about the red dot. Each navy square lands in the opposite corner position.
Check.Turn the whole pattern a half turn: it matches ✓✓2
In plain EnglishCheck by turning the page upside down. The whole shape should look exactly the same as before.
Marking points
M1 Half-turn images
M2 Complete

Reflection

Transformations

Reflecting a shape

Reflect triangle A in the y-axis. Write down the coordinates of the corners of the image.(2 marks)
−5−4−3−2−112345−11234xyA
Reflect each corner.(1, 1) → (−1, 1)(4, 1) → (−4, 1)(1, 3) → (−1, 3)✓1
In plain EnglishThe y-axis is the mirror. Each corner goes the same distance to the other side: the x-coordinate changes sign, the y-coordinate stays the same.
Join them up.−5−4−3−2−112345−11234xyAA′(−1, 1), (−4, 1), (−1, 3)✓2
In plain EnglishCheck: the image is the same size and shape, flipped over.
Marking points
M1 Corners reflected
M2 Image correct
Transformations

A mirror line that isn't an axis

Reflect the shape in the line x = 2.(2 marks)
Count to the mirror line.(0, 1): 2 squares left of x = 2 → (4, 1)(1, 3): 1 square left → (3, 3)(0, 3) → (4, 3)✓1
In plain EnglishFor each corner, count the squares across to the line x = 2, then the same number on the other side.
Draw the image.−11234561234xyx = 2(4, 1), (3, 3), (4, 3)✓2
In plain EnglishPlot the new corners and join them up. Each corner of the image is the same distance from x = 2 as the original, just on the other side.
Marking points
M1 Distances
M2 Image

Translation

Transformations

Translating a shape

Translate square B 4 units right and 1 unit down. Write down the coordinates of the image.(2 marks)
Move every corner the same way.Add 4 to each x-coordinate, take 1 from each y-coordinate✓1(−3, 0) → (1, −1), (−1, 0) → (3, −1)(−1, 2) → (3, 1), (−3, 2) → (1, 1)
In plain EnglishRight is + on x, down is − on y.
The image.−4−3−2−11234−2−1123xyBB′(1, −1), (3, −1), (3, 1), (1, 1)✓2
In plain EnglishThe image is exactly the same shape and the same way round.
Marking points
M1 Rule
M2 Image

Equations of mirror lines

Transformations

Naming the mirror line

Shape P has been reflected to make shape Q. Write down the equation of the mirror line.(1 mark)
−3−2−112345−11234xyPQ
Find the line halfway between.The dashed line goes straight up and down through 1 on the x-axisx = 1✓1
In plain EnglishA vertical line: every point on it has x-coordinate 1.
Marking points
M1 x = 1

0 of 0 complete

Your turn

For the grid questions, tap the squares to shade them, or tap the corners of your image on the coordinate grid (tap again to remove one). Stuck? Tap Need a hint?