064. Line and Rotational Symmetry

Learning Intentions

• To understand what a line of symmetry is
• determine the order of line symmetry for a shape
• To understand what rotational symmetry is
• determine the order of rotational symmetry for a shape

Pre-requisite Summary

• Understand that a shape can be reflected or turned and still be compared with its original position
• Know that congruent parts match exactly in size and shape
• Be able to identify halves of a shape that match under reflection
• Understand that a line of symmetry divides a shape into two matching mirror-image parts
• Know that a full turn is ${360}^{\circ }$
• Understand that rotational symmetry is about turning a shape around a fixed centre
• Be able to count how many times a shape matches itself in one full turn
• Be able to distinguish between reflection symmetry and rotational symmetry

Worked Examples

Worked Example 1

a) Explain what a line of symmetry is.
b) Decide whether a given rectangle has a line of symmetry.
c) State the number of lines of symmetry.

Worked Example 2

For each shape, determine the order of line symmetry:
a) a square
b) an equilateral triangle
c) a regular pentagon

Worked Example 3

a) Explain what rotational symmetry is.
b) Describe what it means for a shape to match itself during a turn.
c) State the order of rotational symmetry of a square.

Worked Example 4

For each shape, determine the order of rotational symmetry:
a) a rectangle
b) an equilateral triangle
c) a regular hexagon

Worked Example 5

Compare the symmetries of these shapes:
a) a non-square rectangle
b) a rhombus
c) a regular octagon
For each, state the order of line symmetry and rotational symmetry.

Worked Example 6

A shape has $4$ lines of symmetry and rotational symmetry of order $4$.
a) Name a possible shape.
b) Explain why it has these symmetries.
c) State the angle of rotation for one turn to the next matching position.

Problems

Problem 1

a) Explain what a line of symmetry is.
b) Decide whether a given isosceles triangle has a line of symmetry.
c) State the number of lines of symmetry.

Problem 2

For each shape, determine the order of line symmetry:
a) a regular hexagon
b) a square
c) a regular octagon

Problem 3

a) Explain what rotational symmetry is.
b) Describe what it means for a shape to match itself during a turn.
c) State the order of rotational symmetry of an equilateral triangle.

Problem 4

For each shape, determine the order of rotational symmetry:
a) a non-square rectangle
b) a regular pentagon
c) a regular octagon

Problem 5

Compare the symmetries of these shapes:
a) a kite
b) a parallelogram
c) a regular hexagon
For each, state the order of line symmetry and rotational symmetry.

Problem 6

A shape has $3$ lines of symmetry and rotational symmetry of order $3$.
a) Name a possible shape.
b) Explain why it has these symmetries.
c) State the angle of rotation for one turn to the next matching position.

Exercises

Understanding and Fluency

1. State whether each statement is true or false:
   a) A line of symmetry divides a shape into two matching halves
   b) A shape with rotational symmetry must match itself after a full turn only
   c) A square has more than one line of symmetry

2. Determine the number of lines of symmetry for each shape:
   a) a square
   b) a rectangle
   c) an equilateral triangle

3. Determine the number of lines of symmetry for each shape:
   a) an isosceles triangle
   b) a regular pentagon
   c) a regular hexagon

4. Determine the order of rotational symmetry for each shape:
   a) a square
   b) a rectangle
   c) an equilateral triangle

5. Determine the order of rotational symmetry for each shape:
   a) a regular pentagon
   b) a regular hexagon
   c) a regular octagon

6. State both the line symmetry and rotational symmetry orders:
   a) a non-square rectangle
   b) a regular triangle
   c) a regular quadrilateral

7. State both the line symmetry and rotational symmetry orders:
   a) a parallelogram
   b) a rhombus
   c) a regular hexagon

8. Find the angle of rotation between matching positions:
   a) rotational symmetry of order $2$
   b) rotational symmetry of order $3$
   c) rotational symmetry of order $4$

Reasoning

9. Explain why a non-square rectangle has rotational symmetry of order $2$.

10. A student says a shape with one line of symmetry must also have rotational symmetry of order $2$. Explain the mistake.

11. Explain why a regular hexagon has the same order of line symmetry and rotational symmetry.

12. A student says a circle has rotational symmetry of order $1$. Explain why this is not a good description.

Problem-solving

13. A logo matches itself after turns of ${90}^{\circ }$. What is its order of rotational symmetry?

14. A tile has $5$ lines of symmetry. Name a regular polygon it could be.

15. A road sign is shaped like an equilateral triangle. State its order of line symmetry and rotational symmetry.

16. A window frame is a non-square rectangle. State its number of lines of symmetry and order of rotational symmetry.

17. A design has rotational symmetry of order $6$. Find the angle of each matching turn.

18. A shape has no line symmetry but does have rotational symmetry of order $2$. Name a possible shape.

Potential Misunderstandings

• Students may think any line through the middle of a shape is a line of symmetry
• Students may confuse the number of lines of symmetry with the number of sides
• Students may think rotational symmetry only means a full turn of ${360}^{\circ }$
• Students may forget that rotational symmetry is about matching before a full turn is completed
• Students may confuse the order of rotational symmetry with the angle of rotation
• Students may assume all regular polygons have only one line of symmetry
• Students may think a shape must have line symmetry if it has rotational symmetry
• Students may count matching turns incorrectly by not including the starting position when considering order

Next: 065. Translations of Shapes and Points