Shape and symmetry
🎯 In this topic you will
- Identify horizontal and vertical lines of symmetry in 2D shapes.
- Sketch the reflection of a 2D shape.
🧠 Key Words
- horizontal
- line of symmetry
- reflection
- symmetry
- vertical
Show Definitions
- horizontal: A direction that goes left to right, parallel to the horizon.
- line of symmetry: A line that splits a shape into two identical mirror-image halves.
- reflection: A transformation that flips a shape across a line (the mirror line) to create a mirror image.
- symmetry: When a shape matches itself after a flip, turn, or other transformation.
- vertical: A direction that goes up and down, at right angles to the horizon.
🪞 Mirror-Image Halves
Asymmetrical shape can be split into two along a line of symmetry. The line of symmetry makes two equal halves that are mirror images of each other.
📏 Two Common Lines
There is a horizontal line of symmetry and a vertical line of symmetry.
🧭 Another Name
The line of symmetry is also called the mirror line.
❓ EXERCISES
$1$. Symmetry means that both sides are the same when split in half along the vertical line or along the horizontal line.
Use this square grid to make two different patterns that are symmetrical.
👀 Show answer
A pattern is symmetrical if the coloured squares match up exactly as mirror images across the line of symmetry.
Example answers (many possible):
Pattern $1$ (vertical symmetry): Colour squares so that for every coloured square on the left of the vertical line, the matching square the same distance on the right is also coloured.
Pattern $2$ (horizontal symmetry): Colour squares so that for every coloured square above the horizontal line, the matching square the same distance below is also coloured.
Any two different patterns are correct as long as they are mirror images across the chosen symmetry line.
$2$. Sketch the reflection.
👀 Show answer
For each part, draw the mirror image of the shape on the other side of the dotted line:
a. Reflect the stepped green line across the horizontal dotted line: every corner and endpoint should be the same perpendicular distance on the opposite side.
b. Reflect the orange curved shape across the vertical dotted line: the curve and the straight edges swap sides at the same distances.
c. Reflect the pink parallelogram across the horizontal dotted line: the whole shape appears below the line with matching distances.
d. Reflect the blue curved-and-straight shape across the vertical dotted line: copy the curve and the straight segment to the left with equal distances from the mirror line.
$3$. Sketch the reflection.
c. Draw two shapes of your own and sketch their reflections. Draw the line of symmetry.
👀 Show answer
a. Reflect the red zig-zag shape across the horizontal dotted line: each vertex is copied to the other side at the same perpendicular distance.
b. Reflect the orange zig-zag shape across the vertical dotted line: each corner is copied to the other side at the same perpendicular distance.
c. Answers will vary. Any two shapes are correct if you draw a mirror line and then draw their reflections so that corresponding points are equal distances from the mirror line.
$4$. Which of the diagrams shows symmetry?
d. Explain what symmetry is.
👀 Show answer
a. Does not show symmetry (the coloured squares do not match as mirror images across the dotted line).
b. Does not show symmetry (the middle coloured squares do not mirror correctly across the dotted line).
c. Shows symmetry (the pattern matches as mirror images across the dotted horizontal line).
d. Symmetry means a shape or pattern can be split along a line so that both halves are identical mirror images; points on one side have matching points the same distance from the line on the other side.
$5$. Colour exactly $7$ squares to show a symmetrical pattern on each grid.
👀 Show answer
Answers will vary. Each grid must have exactly $7$ coloured squares and the pattern must be symmetrical about the dotted line.
a. Use horizontal symmetry: choose $3$ squares above the line and colour their mirror squares below, then colour $1$ square that lies on the symmetry line (so the total is $7$).
b. Use vertical symmetry: choose $3$ squares on the left and colour their mirror squares on the right, then colour $1$ square that lies on the symmetry line.
c. Use diagonal symmetry: choose $3$ squares on one side of the diagonal and colour their mirror squares across the diagonal, then colour $1$ square that lies on the diagonal.
$6$. You have $20$ straws. They are all the same length.
Use squares to draw three different symmetrical patterns that you can make with the straws.
For example:
Each square uses four straws.
Draw the line of symmetry on your pattern.
👀 Show answer
Answers will vary. You must draw $3$ different patterns made from connected squares, and each pattern must have a clear line of symmetry.
One possible set of patterns:
• Pattern $1$: $1$ square (uses $4$ straws) with a vertical symmetry line through the middle.
• Pattern $2$: $3$ squares in a straight row (uses $10$ straws) with a vertical symmetry line through the centre square.
• Pattern $3$: a “T” shape made from $5$ squares (uses $16$ straws) with a vertical symmetry line through the stem of the T.
Any $3$ different designs are correct as long as they are symmetrical and you draw the line of symmetry on each.
🧠 Think like a Mathematician
You will need: A sheet of blank grids.
Task: Using three different colours, explore as many different symmetrical designs as you can by shading whole squares. Mark the lines of symmetry on each pattern.
Method:
- Draw (or print) several blank square grids of the same size.
- Choose three colours to use for shading.
- Shade whole squares only to make a pattern.
- Check that your pattern is symmetrical (both sides match as mirror images).
- Draw the line(s) of symmetry on the grid using a ruler.
- Repeat to create as many different symmetrical designs as you can (try vertical, horizontal, and diagonal symmetry).
👀 show answer
What a correct outcome looks like:
- Each design uses whole shaded squares (no half-squares).
- The design has at least one clear line of symmetry (vertical, horizontal, or diagonal).
- For every shaded square on one side of the symmetry line, there is a matching shaded square the same distance on the other side.
- The symmetry line(s) are drawn clearly on each grid.
There are many correct answers — any pattern is correct as long as it is symmetrical and you mark its line(s) of symmetry.