Lesson 9.6 · Geometry
Line symmetry
Fold a paper heart down the middle and the two halves line up perfectly. Butterflies, snowflakes and many letters of the alphabet have this kind of balance. In math it is called line symmetry.
What is a line of symmetry?
Definition
Line of symmetry
A line of symmetry is a line that divides a figure into two matching halves. If you fold the figure along the line, the two halves fit exactly on top of each other.
A figure that has at least one line of symmetry is symmetric. Each half is a mirror image of the other: every point on one side has a partner the same distance away on the other side.
Counting lines of symmetry
Some figures have more than one line of symmetry. Some have none at all.
| Figure | Lines of symmetry |
|---|---|
| isosceles triangle (only 2 equal sides) | 1 |
| rectangle (not a square) | 2 |
| equilateral triangle | 3 |
| square | 4 |
| circle | as many as you like: every line through the center works |
| scalene triangle | 0 |
The fold test
To check a line, imagine folding along it. If every part of one half lands exactly on the other half, it is a line of symmetry. If anything sticks out, it is not.
Common mistake
A rectangle's diagonal (corner to corner) is not a line of symmetry. Fold a rectangle of paper that way and the corners stick out past each other. Only a square's diagonals are lines of symmetry.
Worked example: Letters
How many lines of symmetry does each letter have: A, B, H, F?
- A: a line straight down the middle. 1 line.
- B: a line straight across the middle. The top bump matches the bottom bump. 1 line.
- H: one line down the middle and one line across the middle. 2 lines.
- F: no fold makes the halves match. 0 lines.
Worked example: Equilateral triangle
How many lines of symmetry does an equilateral triangle have?
All 3 sides are the same length. A line from any corner to the middle of the opposite side folds the triangle onto itself. There are 3 corners, so there are 3 lines of symmetry.
Tip
Shapes with all sides equal and all angles equal have as many lines of symmetry as they have sides: an equilateral triangle has 3, a square has 4, and a regular hexagon has 6.
Practice
How many lines of symmetry does a square have?
Type a number.
How many lines of symmetry does a rectangle that is not a square have?
Type a number.
Which letter has a line of symmetry?
Is the line drawn a line of symmetry for this kite?
How many lines of symmetry does an equilateral triangle have?
Type a number.
Which letter has exactly 2 lines of symmetry?
This hexagon has 6 equal sides and 6 equal angles. How many lines of symmetry does it have?
Type a number.
How many lines of symmetry does this parallelogram have?
Type a number.