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Identifying the symmetry of 2D shapes

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visibility 104update 6 months agobookmarkshare

🎯 In this topic you will

Identify the reflective symmetry of 2D shapes and patterns
Identify the order of rotational symmetry of 2D shapes and patterns

🧠 Key Words

conjecture
full turn
horizontal line
line symmetry
once
rotational symmetry
twice
vertical line

Show Definitions

conjecture: An idea or prediction about a mathematical pattern that may be tested to see if it is true.
full turn: A complete $360^{\circ}$ rotation around a point.
horizontal line: A straight line that runs left to right, parallel to the horizon.
line symmetry: A property where a shape can be folded along a line and both halves match exactly.
once: Describes a rotation that maps a shape onto itself one time during a full turn.
rotational symmetry: When a shape looks the same after being rotated by part of a full turn.
twice: Describes a rotation that maps a shape onto itself two times during a full turn.
vertical line: A straight line that goes up and down, at right angles to the horizon.

✨ Symmetry in 2D Shapes

A 2-dimensional (2D) shape might have line symmetry and it might have rotational symmetry.

This rectangle has two lines of symmetry. One line of symmetry is horizontal. The other line of symmetry is vertical.

You can use dashed lines to show lines of symmetry on a shape.

When you fold a shape along its lines of symmetry, one half of the shape will fit exactly on top of the other half of the shape.

The orange rectangle also has rotational symmetry of order $2$.

The order of rotational symmetry is the number of times a shape looks the same as it is rotated through one full turn.

📘 Worked example

For each of these shapes, write down:
i. the number of lines of symmetry
ii. the order of rotational symmetry

Answer:

a. i. two lines of symmetry
a. ii. order $2$ rotational symmetry

b. i. no lines of symmetry
b. ii. order $2$ rotational symmetry

Shape a. This shape has two diagonal lines of symmetry.

In one full turn, the shape looks the same twice: once after rotating through $180^\circ$ and once again after $360^\circ$.

Shape b. It is not possible to draw any lines of symmetry onto this parallelogram.

In one full turn, the parallelogram looks the same twice: once after rotating through $180^\circ$ and once again after $360^\circ$.

🧠 PROBLEM-SOLVING Strategy

Lines of Symmetry & Rotational Symmetry (2D Shapes & Grid Patterns)

Use these steps to test mirror lines and find the order of rotational symmetry for any shape or tiled pattern.

Choose a candidate mirror line. Try vertical, horizontal, and diagonals through the centre. If the shape is on a grid, prefer lines through grid lines or midpoints.
Reflection test. For every filled square/point, check there is a partner the same distance on the other side of the line. If all match, that line is a line of symmetry. Count them: $0$, $1$, $2$, …
Find the centre of rotation. Usually the geometric centre. On a $3\times3$ grid, it’s the middle cell; on polygons, it’s often the intersection of symmetry axes.
Quarter/half-turn test. Rotate the shape about the centre by $90^\circ$, $180^\circ$, and $270^\circ$. Count how many times it matches its original position in one full turn ($360^\circ$). That count is the order of rotational symmetry.
Use orbit reasoning (grids). Under a $90^\circ$ turn on a $3\times3$ grid, positions fall into orbits: centre $\{1\}$, edge-midpoints $\{4\}$, corners $\{4\}$. A pattern with order $4$ must colour whole orbits.
Common quick checks.
- Rectangle: lines $2$ (vertical horizontal), order $2$.
- Square: lines $4$, order $4$.
- Parallelogram (non-rectangle): lines $0$, order $2$.
- Circle: lines = infinite, order = infinite.
State results clearly. Write “Lines of symmetry: $n$. Order of rotation: $k$.” Add a small sketch with dashed mirror lines or rotation arrows when helpful.

Shape	Lines of symmetry	Order of rotation
Square	$4$	$4$
Rectangle	$2$	$2$
Parallelogram	$0$	$2$
Regular hexagon	$6$	$6$

❓ EXERCISES

1. Copy each of these shapes and draw on them the lines of symmetry.

👀 Show answer

Students should draw the appropriate symmetry lines: some shapes (like the rectangle and circle) have multiple symmetry lines, while others (like the parallelogram) may have none.

2. For each shape in Question 1, write down the order of rotational symmetry.

👀 Show answer

Each shape has a different order of rotational symmetry. For example, a rectangle has order $2$, a circle has infinite order, and a parallelogram (that is not a rectangle) has order $2$.

3. Write down the number of lines of symmetry for each of the following shapes.

👀 Show answer

The number of symmetry lines varies by shape: for example, a regular hexagon has $6$, a semicircle has $1$, and irregular shapes may have none.

4. For each shape in Question 3, write down the order of rotational symmetry.

👀 Show answer

For example, the regular hexagon has rotational symmetry of order $6$, the arrow shape has order $1$, and other irregular shapes may have order $1$ or $2$ depending on their design.

❓ EXERCISES

5. Copy and complete the table to show the symmetry properties of these quadrilaterals.

Shape	Square	Rectangle	Rhombus	Parallelogram	Kite	Trapezium	Isosceles trapezium
Number of lines of symmetry
Order of rotational symmetry

👀 Show answer

Shape	Square	Rectangle	Rhombus	Parallelogram	Kite	Trapezium	Isosceles trapezium
Number of lines of symmetry	$4$	$2$	$2$	$0$	$1$	$0$	$1$
Order of rotational symmetry	$4$	$2$	$2$	$2$	$1$	$1$	$1$

6. For each triangle shown:

i. Write down the number of lines of symmetry.
ii. Write down the order of rotational symmetry.

👀 Show answer

a. Equilateral triangle → Lines of symmetry: $3$, Rotational symmetry: $3$

b. Isosceles triangle → Lines of symmetry: $1$, Rotational symmetry: $1$

c. Scalene triangle → Lines of symmetry: $0$, Rotational symmetry: $1$

d. Right-angled isosceles triangle → Lines of symmetry: $1$, Rotational symmetry: $1$

🧠 Think like a Mathematician

7 a. How many lines of symmetry does a circle have?

b. What is the order of rotation of a circle?

👀 show answer

a: A circle has an infinite number of lines of symmetry, since it can be folded across any diameter.
b: The order of rotational symmetry of a circle is also infinite, because it looks the same after any amount of rotation about its center.

❓ EXERCISES

8 a. For each of these shapes, draw in all the lines of symmetry.

ii.

iii.

b. For each of the shapes in part a, write down the order of rotational symmetry.

👀 Show answer

a. i. Two lines of symmetry — vertical and horizontal.
Order of rotation: $2$

a. ii. Four lines of symmetry — both diagonals, vertical and horizontal.
Order of rotation: $4$

a. iii. No lines of symmetry.
Order of rotation: $1$

❓ EXERCISES

9. Ali and Ritesh are trying to complete this pattern.

The dashed line is a line of symmetry. This is what they draw.

Ali:

Ritesh:

a. Who has drawn the correct diagram?

b. Explain the mistake that the other person has made.

👀 Show answer

a.Ali has the correct diagram.

b. Ritesh has reflected the tiles as if about a different line (or slid/rotated some tiles). For a reflection in the given dashed line, each filled square must have a mirror square the same distance on the other side of the line; Ritesh’s drawing breaks this rule.

10. Copy these patterns onto squared grid paper.

i. Add one blue square to each pattern to make a new pattern that has a line of symmetry.

ii. Draw the line of symmetry onto each of your patterns.

iii. Describe each line of symmetry; that is, is each line a horizontal, vertical or diagonal line of symmetry?

👀 Show answer

Answers will vary. One valid approach for each pattern is to place the added blue square so that every filled square has a mirror partner about a chosen line.

a. Add a square to create a vertical mirror line through the middle column. ($1$ line of symmetry.)
b. Add a square so the pattern is symmetric about the main diagonal. ($1$ line of symmetry.)
c. Add a square to balance left and right for a vertical mirror line. ($1$ line of symmetry.)
d. Add a square to balance top and bottom for a horizontal mirror line. ($1$ line of symmetry.)

For part iii, state the orientation you achieved (vertical, horizontal, or diagonal) for each pattern.

11. Sofia has made this pattern from yellow and blue tiles. She also has two spare blue tiles.

There are eight different ways I can add the two blue tiles to the pattern to make a pattern with only one line of symmetry.

There are two different ways I can add the two blue tiles to the pattern to make a pattern with two lines of symmetry.

There is only one way I can add the two blue tiles to the pattern to make a pattern with four lines of symmetry.

Show that Sofia’s statements are correct. You may join the tiles either side to side or corner to corner.

👀 Show answer

Outline of a counting argument (one valid justification):

Four lines of symmetry. To have symmetry about all four axes of a square grid, both added tiles must be placed in positions invariant under a quarter-turn about the center (i.e., forming a plus-shaped, perfectly balanced pattern). There is exactly $1$ such placement up to the given start pattern, matching the statement.
Two lines of symmetry. Choose a pair of perpendicular axes through the center. The two tiles must be placed symmetrically on one axis and also on the perpendicular axis. This yields exactly $2$ distinct placements (one for each pair of perpendicular axes).
Only one line of symmetry. Choose a single axis (there are $4$ choices through the center). For each chosen axis, the two tiles must be mirror images across that line but not produce symmetry about any other line or a half/quarter-turn. For each axis there are $2$ non-overlapping placements that satisfy this, giving $4 \times 2 = 8$ ways in total.

Thus Sofia’s counts — $8$ ways with one line, $2$ ways with two lines, and $1$ way with four lines — are consistent.

🧠 Think like a Mathematician

12. Read through Question 13a.

a. Without drawing any diagrams, conjecture if the pattern of tiles will be a rectangle, a square or neither. How do you know?

b i. Where in the pattern must you definitely place one red tile? Explain why.

b ii. Now draw the patterns.

👀 Show answer

a: From the information here (without the full text of 13a), a unique shape cannot be confirmed. A useful test:
- If intended side lengths are equal and both directions are balanced, conjecture square.
- If opposite sides are equal in pairs but the two side lengths differ, conjecture rectangle.
- If neither of the above balancing conditions holds, conjecture neither.
State which balances (row/column counts or symmetry cues) lead to your choice.
b i: A tile that guarantees symmetry must lie on the line of symmetry.
- With one mirror line → place the red tile somewhere on that line.
- With two perpendicular mirror lines → place it at their intersection (the center).
- With rotational symmetry only (no mirror line) → place it at the center of rotation.
This ensures every tile has a symmetric partner (or maps to itself under the rotation).
b ii: Draw one or more patterns consistent with your conjecture. Mark the symmetry line(s) or rotation center; check that reflected/rotated positions match exactly.

❓ EXERCISES

13. Song has five red tiles and four white tiles.

a. Draw two different ways that Song could arrange these tiles so that he has a shape with an order of rotational symmetry of $4$.

b. For each of the patterns you drew in part a, how many lines of symmetry do your patterns of tiles have?

👀 Show answer

Key idea: A pattern with rotational symmetry of order $4$ must be invariant under a $90^\circ$ turn. On a $3\times 3$ grid the positions form orbits: center ($1$), edge-midpoints ($4$), corners ($4$). With $5$ red and $4$ white, the red tiles must be the center plus one entire orbit of size $4$.

a. Two different correct arrangements (both have order $4$):

Plus pattern: Put a red tile at the center and red tiles at the four edge-midpoints. The four corner tiles are white.
Diamond pattern: Put a red tile at the center and red tiles at the four corners. The four edge-midpoint tiles are white.

b. Each of the above patterns has $4$ lines of symmetry (vertical, horizontal, and the two diagonals).

📘 What we've learned

We identified and drew lines of symmetry for 2D shapes and patterns.
We defined and calculated the order of rotational symmetry as the number of times a shape fits onto itself in a full turn ($360^\circ$).
We practiced testing symmetry using $90^\circ$, $180^\circ$, and $270^\circ$ turns.
We learned how orbits (center, corners, edge-midpoints) explain rotational symmetry on a $3 \times 3$ grid.
We compared symmetry in familiar shapes: square ($4$ lines, order $4$), rectangle ($2$ lines, order $2$), parallelogram ($0$ lines, order $2$).
We applied these strategies to tiling problems and explained why certain arrangements have the required symmetry.

Identifying the symmetry of 2D shapes

🎯 In this topic you will

🧠 Key Words

✨ Symmetry in 2D Shapes

🧠 PROBLEM-SOLVING Strategy

Lines of Symmetry & Rotational Symmetry (2D Shapes & Grid Patterns)

❓ EXERCISES

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

📘 What we've learned

Related Past Papers

Related Tutorials

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