Line Symmetry and Rotational Symmetry
A figure has symmetry when a transformation maps it onto itself. This lesson explores two types: line (reflective) symmetry and rotational symmetry.
Introduction
You've learned translations, reflections, and rotations as individual moves. Now ask: what if the figure looks exactly the sameafter a transformation? That's symmetry — the fingerprint of beauty in geometry, nature, and design.
Past Knowledge
Reflections (4.1.4–4.1.5). Rotations (4.1.6). Coordinate rules.
Today's Goal
Identify lines of symmetry and determine the order & angle of rotational symmetry.
Future Success
Compositions (4.2.2), tessellations, and symmetry groups in advanced math.
Key Concepts
Line Symmetry (Reflective Symmetry)
A figure has line symmetry if there exists a line (called the line of symmetry or axis of symmetry) such that reflecting the figure across that line maps it onto itself.
Each half of the figure is a mirror image of the other.
Rotational Symmetry
A figure has rotational symmetry if there exists a rotation of less than 360° that maps the figure onto itself.
Order of symmetry = the number of positions in which the figure looks the same during a full 360° turn.
Angle of symmetry =
Lines of Symmetry for Regular Polygons
| Regular Polygon | Lines of Symmetry | Rotational Order | Angle |
|---|---|---|---|
| Equilateral triangle | 3 | 3 | 120° |
| Square | 4 | 4 | 90° |
| Regular pentagon | 5 | 5 | 72° |
| Regular hexagon | 6 | 6 | 60° |
| Regular -gon |
Worked Examples
Line Symmetry of a Rectangle
A rectangle has vertices , , , . Identify all lines of symmetry.
Reflect across the x-axis: .
, . The rectangle maps to itself. ✓
Reflect across the y-axis: .
, . Maps to itself. ✓
2 lines of symmetry — the x-axis and the y-axis.
Rotational Symmetry of a Square
A square has vertices , , , . Find the order and angle of rotational symmetry.
Rotate 90° CCW: , . Maps to itself. ✓
The square maps to itself at 90°, 180°, 270°, and 360°.
Order 4, angle of symmetry = .
Symmetry of a Parallelogram
A parallelogram has vertices , , , . Does it have line symmetry? Rotational symmetry?
Line symmetry: No axis or diagonal reflection maps the parallelogram onto itself (unless it's a rectangle or rhombus). A generic parallelogram has no lines of symmetry.
Rotational symmetry: Rotate 180° about the center :
✓, ✓.
No line symmetry. Rotational symmetry of order 2 (angle = 180°).
Common Pitfalls
Confusing “Lines of Symmetry” with “Diagonals”
The diagonals of a parallelogram are not lines of symmetry. A line of symmetry must map the figure onto itself — test it by reflecting.
Counting 360° as Rotational Symmetry
Every figure maps to itself under a 360° rotation, so we don't count it. Rotational symmetry requires a rotation of less than 360°.
Assuming All Regular Figures Have the Same Symmetry
An equilateral triangle has order-3 rotational symmetry (120°), while a square has order-4 (90°). The number of sides determines the symmetry.
Real-Life Applications
Nature — Snowflakes & Flowers
Snowflakes exhibit 6-fold rotational symmetry (order 6, angle 60°) and 6 lines of symmetry. Many flowers display 5-fold symmetry. Recognizing these patterns helps biologists classify organisms.
Logos & Brand Design
Designers use symmetry to create visually balanced and memorable logos. The Mercedes-Benz logo has 3-fold rotational symmetry; the Target bullseye has infinite rotational symmetry (a circle).
Practice Quiz
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