IMOClass 6 › Symmetry

Symmetry

Lines of Symmetry

A shape has a line of symmetry if a line divides it into two halves that match exactly, like a mirror. A square has 4 lines of symmetry, a rectangle has 2, an equilateral triangle has 3 and a circle has many.

Letters can be symmetric too — A has a vertical line of symmetry.

Example 1: How many lines of symmetry does a square have?
4 lines of symmetry.
Example 2: What kind of line of symmetry does the letter A have?
A vertical line of symmetry.
Quick recap
  • A line of symmetry gives two matching halves.
  • Square 4, rectangle 2, equilateral triangle 3, circle many.
✓ Quick check
How many lines of symmetry does a rectangle have?
A rectangle has 2 lines of symmetry.
How many lines of symmetry does an equilateral triangle have?
An equilateral triangle has 3 lines of symmetry.

Reflection & Mirror Symmetry

In reflection, a shape and its mirror image are the same distance from the mirror line and face opposite ways. A shape has mirror symmetry if one half is the mirror image of the other.

Letters like H, I, O and X have both a vertical and a horizontal line of symmetry.

Example 1: Which letter has a vertical line of symmetry: A, F or G?
A — its left and right halves match.
Example 2: Name a letter with both vertical and horizontal symmetry.
H (also I, O, X).
Quick recap
  • Reflection flips a shape across a mirror line.
  • H, I, O, X have both vertical and horizontal symmetry.
✓ Quick check
Which letter has a vertical line of symmetry?
The letter A has a vertical line of symmetry.
Which letter has both horizontal and vertical symmetry?
H has both a vertical and a horizontal line of symmetry.

Rotational Symmetry

A shape has rotational symmetry if it looks the same after turning by some angle less than a full turn. The order is how many times it matches in one full turn. A square has order 4 (turning 90° each time); a rectangle has order 2 (180°); an equilateral triangle has order 3 (120°).

Example 1: What is the order of rotational symmetry of a square?
4 (it matches every 90°).
Example 2: What angle do you turn a square so it looks the same?
90°.
Quick recap
  • Order = number of matches in one full turn.
  • Square order 4 (90°), rectangle order 2 (180°), equilateral triangle order 3 (120°).
✓ Quick check
What is the order of rotational symmetry of a rectangle?
A rectangle matches twice in a full turn, so order 2.
What is the order of rotational symmetry of an equilateral triangle?
It matches three times in a full turn, so order 3.
Ready to test this chapter?
Take the Chapter Test →