Symmetry booklet
Symmetry booklet
- Unit 1: Numbers
- Unit 2: Geometry and measure
- Unit 3 : Statistics and probability
🎯 In this topic you will
- Improve your understanding of symmetry in two-dimensional (2D) shapes.
- Identify and find all lines of symmetry in 2D shapes and patterns.
🧠 Key Words
- horizontal line of symmetry
- vertical line of symmetry
- line of symmetry
Show Definitions
- horizontal line of symmetry: A horizontal line that divides a shape into two identical halves, where the top half is a mirror image of the bottom half.
- vertical line of symmetry: A vertical line that divides a shape into two identical halves, where the left half is a mirror image of the right half.
- line of symmetry: A line that splits a shape into two equal and matching parts that can be folded onto each other exactly.
Beauty and Balance in Symmetry
S ymmetrical patterns are usually beautiful and fascinating. You can see symmetry all around you in nature and in art and design. Learning about symmetry helps you to notice similarity, difference, and balance, which are important ideas in all areas of mathematics.
❓ EXERCISES
$1$. How many lines of symmetry does each pattern have?
Use a mirror to check for lines of symmetry.
Check for a vertical line, a horizontal line and the two diagonal lines.
👀 Show answer
a. The pattern has $2$ lines of symmetry (one vertical and one horizontal).
b. The pattern has $4$ lines of symmetry (vertical, horizontal, and both diagonals).
c. The pattern has $1$ line of symmetry (a horizontal line).
d. The pattern has $4$ lines of symmetry (vertical, horizontal, and both diagonals).
e. The pattern has $0$ lines of symmetry.
f. The pattern has $2$ lines of symmetry (both diagonals).
g. The pattern has $4$ lines of symmetry (vertical, horizontal, and both diagonals).
h. The pattern has $0$ lines of symmetry.
$2$. How many lines of symmetry does each pattern have?
👀 Show answer
a. The pattern has $2$ lines of symmetry (one vertical and one horizontal).
b. The pattern has $0$ lines of symmetry.
c. The pattern has $4$ lines of symmetry (vertical, horizontal, and both diagonals).
d. The pattern has $0$ lines of symmetry.
e. The pattern has $0$ lines of symmetry.
f. The pattern has $0$ lines of symmetry.
g. The pattern has $4$ lines of symmetry (vertical, horizontal, and both diagonals).
h. The pattern has $1$ line of symmetry (a vertical line).
❓ EXERCISES
3. Trace and cut out these shapes. How many lines of symmetry do the shapes have?
👀 Show answer
a. $1$ line of symmetry.
b. $1$ line of symmetry.
c. $3$ lines of symmetry.
d. $0$ lines of symmetry.
e. $4$ lines of symmetry.
f. $5$ lines of symmetry.
g. $0$ lines of symmetry.
h. $2$ lines of symmetry.
4. A parallelogram is characterised as a $4$-sided polygon with two pairs of parallel sides. Which of these parallelograms have diagonal lines of symmetry? Test your conjectures.
The parallelograms that have lines of symmetry have a special property.
Measure the lengths of the sides of the parallelograms to find out the special property.
Copy and complete the following generalisation:
The parallelograms that have diagonal lines of symmetry all have …
👀 Show answer
The parallelograms with diagonal lines of symmetry are those where all sides are equal in length.
These shapes are rhombuses (including squares).
5. All of these shapes have four sides.
a. Do they have the same number of lines of symmetry?
b. Which shapes have the fewest lines of symmetry?
c. Which shape has the most lines of symmetry?
👀 Show answer
a. No, they do not all have the same number of lines of symmetry.
b. The shapes with the fewest lines of symmetry are the irregular quadrilaterals.
c. The square has the most lines of symmetry.
🧠 Think like a Mathematician
Investigate the number of lines of symmetry in these regular polygons.
a. Trace the shapes and draw on their lines of symmetry.
You could use a mirror or you could fold them to find the lines of symmetry.
b. Copy and complete this table to record the characteristics of each shape.
| Shape | Name | Sides | Vertices | Lines of symmetry |
|---|---|---|---|---|
| A | Equilateral triangle | $3$ | $3$ | $3$ |
| B | Square | $4$ | $4$ | $4$ |
| C | Regular pentagon | $5$ | $5$ | $5$ |
| D | Regular hexagon | $6$ | $6$ | $6$ |
| E | Regular heptagon | $7$ | $7$ | $7$ |
| F | Regular octagon | $8$ | $8$ | $8$ |
Show Answers
For a regular polygon, the number of lines of symmetry is always equal to the number of sides.
This happens because all sides and angles are equal, so the shape can be folded onto itself in the same number of ways.
💡 Quick Math Tip
Regular polygons and symmetry: For any regular polygon, the number of lines of symmetry is always the same as the number of sides, because all sides and angles are equal.