Symmetry
🎯 In this topic you will
- Explore the symmetry in triangles
- Explore the symmetry in patterns
- Create patterns with lines of symmetry
🧠 Key Words
- line of symmetry
- symmetrical
Show Definitions
- line of symmetry: A line that divides a shape into two identical parts that are mirror images of each other.
- symmetrical: Describes a shape that can be divided into matching halves by at least one line of symmetry.
Symmetry in the World Around Us
S ymmetry is all around you in nature and in art and design. In this section you will learn how to create symmetrical patterns. Learning about symmetry helps you to notice similarity, difference and balance, which is important to all parts of mathematics.
💡 Quick Math Tip
Spotting Symmetry Helps You Compare Shapes: When you look for lines of symmetry, it becomes easier to notice similarity, difference, and balance — skills that are useful throughout mathematics.
❓ EXERCISES
1. How many lines of symmetry does each triangle have?
a. Triangle a
b. Triangle b
c. Triangle c
d. Triangle d
e. Triangle e
f. Triangle f
👀 Show answer
a. $0$ lines of symmetry
b. $0$ lines of symmetry
c. $1$ line of symmetry (isosceles)
d. $0$ lines of symmetry
e. $0$ lines of symmetry
f. $1$ line of symmetry (isosceles)
2. These tiles have reflective symmetry in their shape and in their patterns. How many lines of symmetry does each pattern have?
a. Pattern a
b. Pattern b
c. Pattern c
d. Pattern d
e. Pattern e
👀 Show answer
a. $6$ lines of symmetry
b. $1$ line of symmetry
c. $4$ lines of symmetry (square-based pattern)
d. $0$ lines of symmetry
e. $10$ lines of symmetry (decagonal rotational design)
3. Look at this coloured pattern.
What colour can you colour the square at * to make a pattern with exactly:
a. one line of symmetry?
b. two lines of symmetry?
c. more than two lines of symmetry?
👀 Show answer
4. Look at this tile pattern.
If the pattern is reflected in the two red mirror lines, what colour will these squares be?
A B
C D
E F
G
👀 Show answer
5. Copy and complete this pattern by reflecting the shapes over both mirror lines until there are three shapes in each quadrant.
👀 Show answer
🧠 Think like a Mathematician
This is a square grid of 9 squares.
Use one coloured pen or pencil and squared paper.
How many patterns with at least 1 line of symmetry can you make by shading 4 squares?
Try not to make patterns that look the same from different directions.
For example, these two patterns are the same:
Now your investigation continues:
Choose one thing to change and write your own question to investigate.
How many patterns with ____ lines of symmetry can be made by shading ____ squares?
You could change:
- the number of squares in the grid
- the number of shaded squares
- the minimum number of lines of symmetry
Investigate your question and write your solution.
- You are specialising when you make a pattern and test it to check if it has at least one line of symmetry.
- You are conjecturing when you write your own question to investigate.
- You are improving when you reflect on your investigation and consider how you could improve your approach.
Follow-up Questions
Show Answers
- 1: There are only a small number of possible symmetric arrangements, because a 3×3 grid supports only vertical, horizontal, and diagonal reflection. Careful counting shows that exactly 3 distinct patterns of shading 4 squares have at least one line of symmetry.
- 2: Changing the number of shaded squares changes both the total number of possible patterns and how many of them can be symmetric. Some numbers of shaded squares allow many symmetric arrangements, while others allow very few or none.
- 3: A reliable method is to generate patterns systematically: start with all patterns with vertical symmetry, then those with horizontal symmetry, then diagonal symmetry. At each stage eliminate rotated or reflected duplicates.