Symmetry - Tenth Grade Geometry
Introduction to Symmetry
Symmetry: A figure has symmetry if it can be transformed onto itself
Two Main Types:
1. Line Symmetry (Reflection Symmetry) - Mirror image across a line
2. Rotational Symmetry - Looks the same after rotation
Key Concept: Symmetry is about balance, matching parts, and repetition of patterns
Found in: Nature, art, architecture, biology, and mathematics
Two Main Types:
1. Line Symmetry (Reflection Symmetry) - Mirror image across a line
2. Rotational Symmetry - Looks the same after rotation
Key Concept: Symmetry is about balance, matching parts, and repetition of patterns
Found in: Nature, art, architecture, biology, and mathematics
1. Line Symmetry (Reflection Symmetry)
Line Symmetry: A figure has line symmetry if it can be folded along a line so that both halves match exactly
Line of Symmetry: The line where you could place a mirror and see the complete figure
Also Called: Mirror symmetry, reflection symmetry, bilateral symmetry
Key Property: Each point on one side has a corresponding point on the other side, equidistant from the line
Line of Symmetry: The line where you could place a mirror and see the complete figure
Also Called: Mirror symmetry, reflection symmetry, bilateral symmetry
Key Property: Each point on one side has a corresponding point on the other side, equidistant from the line
Properties of Line Symmetry:
1. Mirror Image:
• One half is the exact reflection of the other half
• If you fold along the line, both sides match perfectly
2. Perpendicular Distance:
• Every point and its reflection are the same distance from the line of symmetry
• The line of symmetry is the perpendicular bisector of the segment connecting each point to its image
3. Direction of Line:
• Can be vertical, horizontal, or diagonal
• Can be in any orientation
4. Number of Lines:
• A figure can have zero, one, or multiple lines of symmetry
• Some figures have infinite lines of symmetry (circle)
1. Mirror Image:
• One half is the exact reflection of the other half
• If you fold along the line, both sides match perfectly
2. Perpendicular Distance:
• Every point and its reflection are the same distance from the line of symmetry
• The line of symmetry is the perpendicular bisector of the segment connecting each point to its image
3. Direction of Line:
• Can be vertical, horizontal, or diagonal
• Can be in any orientation
4. Number of Lines:
• A figure can have zero, one, or multiple lines of symmetry
• Some figures have infinite lines of symmetry (circle)
Types of Line Symmetry
1. Vertical Line of Symmetry:
• Line runs up and down (vertical)
• Left side mirrors the right side
• Example: Letter A, letter M, heart shape
2. Horizontal Line of Symmetry:
• Line runs left to right (horizontal)
• Top mirrors the bottom
• Example: Letter B, letter D, rectangle
3. Diagonal Line of Symmetry:
• Line runs at an angle
• Often seen in squares and regular polygons
• Example: Square has 2 diagonal lines of symmetry
4. Multiple Lines:
• Some figures have more than one line of symmetry
• Regular polygons have multiple lines
• Circle has infinite lines of symmetry
• Line runs up and down (vertical)
• Left side mirrors the right side
• Example: Letter A, letter M, heart shape
2. Horizontal Line of Symmetry:
• Line runs left to right (horizontal)
• Top mirrors the bottom
• Example: Letter B, letter D, rectangle
3. Diagonal Line of Symmetry:
• Line runs at an angle
• Often seen in squares and regular polygons
• Example: Square has 2 diagonal lines of symmetry
4. Multiple Lines:
• Some figures have more than one line of symmetry
• Regular polygons have multiple lines
• Circle has infinite lines of symmetry
Example 1: Identify line symmetry
Which letters have vertical line symmetry?
A, H, I, M, O, T, U, V, W, X, Y
Which letters have horizontal line symmetry?
B, C, D, H, I, O, X
Which letters have both?
H, I, O, X
Which letters have vertical line symmetry?
A, H, I, M, O, T, U, V, W, X, Y
Which letters have horizontal line symmetry?
B, C, D, H, I, O, X
Which letters have both?
H, I, O, X
Example 2: Real-world symmetry
Objects with line symmetry:
• Butterfly wings (vertical)
• Human face (vertical)
• Snowflakes (multiple lines)
• Playing cards (vertical or horizontal)
• Leaves (sometimes vertical)
• Buildings and architecture
Objects with line symmetry:
• Butterfly wings (vertical)
• Human face (vertical)
• Snowflakes (multiple lines)
• Playing cards (vertical or horizontal)
• Leaves (sometimes vertical)
• Buildings and architecture
2. Rotational Symmetry
Rotational Symmetry: A figure has rotational symmetry if it looks the same after being rotated by less than 360° around a central point
Center of Rotation: The fixed point around which the figure rotates
Angle of Rotation: The smallest angle through which the figure can be rotated to look the same
Order of Rotational Symmetry: The number of times a figure matches itself during one complete rotation (360°)
Center of Rotation: The fixed point around which the figure rotates
Angle of Rotation: The smallest angle through which the figure can be rotated to look the same
Order of Rotational Symmetry: The number of times a figure matches itself during one complete rotation (360°)
Rotational Symmetry Formulas:
1. Order of Rotational Symmetry:
$$\text{Order} = \frac{360°}{\text{Angle of Rotation}}$$
2. Angle of Rotation:
$$\text{Angle of Rotation} = \frac{360°}{\text{Order}}$$
Where:
• Order = number of times figure looks the same in 360° rotation
• Angle of rotation = smallest angle for figure to match itself
Important:
• All figures have rotational symmetry of order 1 (at 360°)
• True rotational symmetry means order ≥ 2
1. Order of Rotational Symmetry:
$$\text{Order} = \frac{360°}{\text{Angle of Rotation}}$$
2. Angle of Rotation:
$$\text{Angle of Rotation} = \frac{360°}{\text{Order}}$$
Where:
• Order = number of times figure looks the same in 360° rotation
• Angle of rotation = smallest angle for figure to match itself
Important:
• All figures have rotational symmetry of order 1 (at 360°)
• True rotational symmetry means order ≥ 2
Key Properties of Rotational Symmetry:
1. Order:
• Order 1: Only looks the same at 360° (no rotational symmetry)
• Order 2: Looks the same at 180° and 360°
• Order 3: Looks the same at 120°, 240°, and 360°
• Order 4: Looks the same at 90°, 180°, 270°, and 360°
• Order n: Looks the same n times during full rotation
2. Regular Polygons:
• A regular n-sided polygon has order n rotational symmetry
• Angle of rotation = $\frac{360°}{n}$
3. Shapes with High Symmetry:
• Circle: Infinite order (any angle works)
• Regular polygons: Order equals number of sides
1. Order:
• Order 1: Only looks the same at 360° (no rotational symmetry)
• Order 2: Looks the same at 180° and 360°
• Order 3: Looks the same at 120°, 240°, and 360°
• Order 4: Looks the same at 90°, 180°, 270°, and 360°
• Order n: Looks the same n times during full rotation
2. Regular Polygons:
• A regular n-sided polygon has order n rotational symmetry
• Angle of rotation = $\frac{360°}{n}$
3. Shapes with High Symmetry:
• Circle: Infinite order (any angle works)
• Regular polygons: Order equals number of sides
Example 1: Find order of rotational symmetry
Square:
Looks the same at: 90°, 180°, 270°, 360°
Order = 4
Equilateral Triangle:
Looks the same at: 120°, 240°, 360°
Order = 3
Regular Pentagon:
Looks the same at: 72°, 144°, 216°, 288°, 360°
Order = 5
Regular Hexagon:
Looks the same at: 60°, 120°, 180°, 240°, 300°, 360°
Order = 6
Square:
Looks the same at: 90°, 180°, 270°, 360°
Order = 4
Equilateral Triangle:
Looks the same at: 120°, 240°, 360°
Order = 3
Regular Pentagon:
Looks the same at: 72°, 144°, 216°, 288°, 360°
Order = 5
Regular Hexagon:
Looks the same at: 60°, 120°, 180°, 240°, 300°, 360°
Order = 6
Example 2: Find angle of rotation
If order = 4, find angle of rotation:
$$\text{Angle} = \frac{360°}{4} = 90°$$
Answer: The figure rotates 90° to match itself
If order = 4, find angle of rotation:
$$\text{Angle} = \frac{360°}{4} = 90°$$
Answer: The figure rotates 90° to match itself
Example 3: Find order given angle
If angle of rotation = 45°, find order:
$$\text{Order} = \frac{360°}{45°} = 8$$
Answer: Order of rotational symmetry is 8
If angle of rotation = 45°, find order:
$$\text{Order} = \frac{360°}{45°} = 8$$
Answer: Order of rotational symmetry is 8
Example 4: Common shapes
Rectangle (not square):
Looks the same at 180° and 360°
Order = 2
Parallelogram:
Looks the same at 180° and 360°
Order = 2
Isosceles Triangle (not equilateral):
Only looks the same at 360°
Order = 1 (no rotational symmetry)
Rectangle (not square):
Looks the same at 180° and 360°
Order = 2
Parallelogram:
Looks the same at 180° and 360°
Order = 2
Isosceles Triangle (not equilateral):
Only looks the same at 360°
Order = 1 (no rotational symmetry)
3. Draw Lines of Symmetry
Drawing Lines of Symmetry: Identifying and marking all possible lines that divide a figure into matching halves
Key Skill: Visualizing where to fold the figure so both sides match
Tools: Can use tracing paper, folding, or mirrors to verify
Key Skill: Visualizing where to fold the figure so both sides match
Tools: Can use tracing paper, folding, or mirrors to verify
Steps to Draw Lines of Symmetry:
Step 1: Analyze the Figure
• Look for matching parts
• Identify the center or axis of balance
• Determine if it's a regular polygon
Step 2: Check Vertical Line
• Draw a vertical line through the center
• Check if left and right sides match
• If yes, this is a line of symmetry
Step 3: Check Horizontal Line
• Draw a horizontal line through the center
• Check if top and bottom match
• If yes, this is a line of symmetry
Step 4: Check Diagonal Lines
• For squares, rectangles, and regular polygons
• Draw lines through opposite vertices or midpoints
• Check if both halves match
Step 5: For Regular Polygons
• Draw lines from each vertex to the midpoint of the opposite side
• Or through opposite vertices (if even number of sides)
• Or from vertex perpendicular to opposite side (if odd sides)
Step 6: Verify
• Imagine folding along each line
• Both halves should match exactly
Step 1: Analyze the Figure
• Look for matching parts
• Identify the center or axis of balance
• Determine if it's a regular polygon
Step 2: Check Vertical Line
• Draw a vertical line through the center
• Check if left and right sides match
• If yes, this is a line of symmetry
Step 3: Check Horizontal Line
• Draw a horizontal line through the center
• Check if top and bottom match
• If yes, this is a line of symmetry
Step 4: Check Diagonal Lines
• For squares, rectangles, and regular polygons
• Draw lines through opposite vertices or midpoints
• Check if both halves match
Step 5: For Regular Polygons
• Draw lines from each vertex to the midpoint of the opposite side
• Or through opposite vertices (if even number of sides)
• Or from vertex perpendicular to opposite side (if odd sides)
Step 6: Verify
• Imagine folding along each line
• Both halves should match exactly
Example 1: Square
Lines of symmetry in a square:
1. Vertical line through center
2. Horizontal line through center
3. Diagonal from top-left to bottom-right
4. Diagonal from top-right to bottom-left
Total: 4 lines of symmetry
Lines of symmetry in a square:
1. Vertical line through center
2. Horizontal line through center
3. Diagonal from top-left to bottom-right
4. Diagonal from top-right to bottom-left
Total: 4 lines of symmetry
Example 2: Rectangle (not square)
Lines of symmetry in a rectangle:
1. Vertical line through center (dividing left and right)
2. Horizontal line through center (dividing top and bottom)
Note: Diagonals are NOT lines of symmetry in a rectangle
Total: 2 lines of symmetry
Lines of symmetry in a rectangle:
1. Vertical line through center (dividing left and right)
2. Horizontal line through center (dividing top and bottom)
Note: Diagonals are NOT lines of symmetry in a rectangle
Total: 2 lines of symmetry
Example 3: Equilateral triangle
Lines of symmetry:
1. From top vertex to midpoint of bottom side
2. From bottom-left vertex to midpoint of opposite side
3. From bottom-right vertex to midpoint of opposite side
These are also the three medians/altitudes/angle bisectors
Total: 3 lines of symmetry
Lines of symmetry:
1. From top vertex to midpoint of bottom side
2. From bottom-left vertex to midpoint of opposite side
3. From bottom-right vertex to midpoint of opposite side
These are also the three medians/altitudes/angle bisectors
Total: 3 lines of symmetry
Example 4: Regular hexagon
Lines of symmetry:
• 3 lines through opposite vertices
• 3 lines through midpoints of opposite sides
Total: 6 lines of symmetry
Lines of symmetry:
• 3 lines through opposite vertices
• 3 lines through midpoints of opposite sides
Total: 6 lines of symmetry
4. Count Lines of Symmetry
Counting Lines of Symmetry: Determining the total number of lines that divide a figure into matching halves
Key Rule: Regular n-sided polygon has exactly n lines of symmetry
Special Cases: Circles (infinite), irregular shapes (varies)
Key Rule: Regular n-sided polygon has exactly n lines of symmetry
Special Cases: Circles (infinite), irregular shapes (varies)
Formula for Regular Polygons:
For a regular polygon with $n$ sides:
$$\text{Number of lines of symmetry} = n$$
Examples:
• Equilateral triangle (n = 3): 3 lines
• Square (n = 4): 4 lines
• Regular pentagon (n = 5): 5 lines
• Regular hexagon (n = 6): 6 lines
• Regular n-gon: n lines
Circle: Infinite lines of symmetry (any diameter is a line of symmetry)
For a regular polygon with $n$ sides:
$$\text{Number of lines of symmetry} = n$$
Examples:
• Equilateral triangle (n = 3): 3 lines
• Square (n = 4): 4 lines
• Regular pentagon (n = 5): 5 lines
• Regular hexagon (n = 6): 6 lines
• Regular n-gon: n lines
Circle: Infinite lines of symmetry (any diameter is a line of symmetry)
Common Shapes and Their Lines of Symmetry:
Triangles:
• Equilateral: 3 lines
• Isosceles: 1 line (through vertex angle perpendicular to base)
• Scalene: 0 lines
Quadrilaterals:
• Square: 4 lines (2 through sides, 2 through diagonals)
• Rectangle: 2 lines (through midpoints of opposite sides)
• Rhombus: 2 lines (through diagonals)
• Parallelogram: 0 lines
• Trapezoid (general): 0 lines
• Isosceles trapezoid: 1 line (perpendicular bisector of parallel sides)
• Kite: 1 line (through two vertices)
Regular Polygons:
• Pentagon: 5 lines
• Hexagon: 6 lines
• Heptagon: 7 lines
• Octagon: 8 lines
• Decagon: 10 lines
• n-gon: n lines
Other Shapes:
• Circle: Infinite lines
• Ellipse: 2 lines (major and minor axes)
• Semicircle: 1 line (perpendicular to diameter at midpoint)
Triangles:
• Equilateral: 3 lines
• Isosceles: 1 line (through vertex angle perpendicular to base)
• Scalene: 0 lines
Quadrilaterals:
• Square: 4 lines (2 through sides, 2 through diagonals)
• Rectangle: 2 lines (through midpoints of opposite sides)
• Rhombus: 2 lines (through diagonals)
• Parallelogram: 0 lines
• Trapezoid (general): 0 lines
• Isosceles trapezoid: 1 line (perpendicular bisector of parallel sides)
• Kite: 1 line (through two vertices)
Regular Polygons:
• Pentagon: 5 lines
• Hexagon: 6 lines
• Heptagon: 7 lines
• Octagon: 8 lines
• Decagon: 10 lines
• n-gon: n lines
Other Shapes:
• Circle: Infinite lines
• Ellipse: 2 lines (major and minor axes)
• Semicircle: 1 line (perpendicular to diameter at midpoint)
Example 1: Count lines for regular polygons
Regular octagon:
Number of sides n = 8
Number of lines of symmetry = 8
Regular decagon:
Number of sides n = 10
Number of lines of symmetry = 10
Regular 20-gon:
Number of sides n = 20
Number of lines of symmetry = 20
Regular octagon:
Number of sides n = 8
Number of lines of symmetry = 8
Regular decagon:
Number of sides n = 10
Number of lines of symmetry = 10
Regular 20-gon:
Number of sides n = 20
Number of lines of symmetry = 20
Example 2: Count lines for letters
Letter A: 1 line (vertical through center)
Letter B: 1 line (horizontal through center)
Letter C: 1 line (horizontal through center)
Letter H: 2 lines (vertical and horizontal)
Letter O: Infinite lines (if perfect circle) or 2 lines (if oval)
Letter X: 2 lines (vertical and horizontal) or 4 lines (including diagonals if perfectly symmetrical)
Letter Z: 0 lines (but has rotational symmetry of order 2)
Letter A: 1 line (vertical through center)
Letter B: 1 line (horizontal through center)
Letter C: 1 line (horizontal through center)
Letter H: 2 lines (vertical and horizontal)
Letter O: Infinite lines (if perfect circle) or 2 lines (if oval)
Letter X: 2 lines (vertical and horizontal) or 4 lines (including diagonals if perfectly symmetrical)
Letter Z: 0 lines (but has rotational symmetry of order 2)
Example 3: Mixed problems
How many lines of symmetry does a regular hexagon have?
Answer: 6 lines
Which quadrilateral has exactly 1 line of symmetry?
Answer: Isosceles trapezoid or kite
Which shape has the most lines of symmetry?
Answer: Circle (infinite lines)
How many lines of symmetry does a regular hexagon have?
Answer: 6 lines
Which quadrilateral has exactly 1 line of symmetry?
Answer: Isosceles trapezoid or kite
Which shape has the most lines of symmetry?
Answer: Circle (infinite lines)
Quick Memory Trick:
For Regular Polygons:
"Sides = Symmetry lines"
A regular polygon with n sides has n lines of symmetry
Also remember:
Regular polygon with n sides also has rotational symmetry of order n
This means: Lines of symmetry = Order of rotational symmetry = Number of sides
For Regular Polygons:
"Sides = Symmetry lines"
A regular polygon with n sides has n lines of symmetry
Also remember:
Regular polygon with n sides also has rotational symmetry of order n
This means: Lines of symmetry = Order of rotational symmetry = Number of sides
Symmetry in Common Shapes
| Shape | Lines of Symmetry | Order of Rotational Symmetry | Angle of Rotation |
|---|---|---|---|
| Equilateral Triangle | 3 | 3 | 120° |
| Isosceles Triangle | 1 | 1 (none) | 360° |
| Scalene Triangle | 0 | 1 (none) | 360° |
| Square | 4 | 4 | 90° |
| Rectangle | 2 | 2 | 180° |
| Rhombus | 2 | 2 | 180° |
| Parallelogram | 0 | 2 | 180° |
| Isosceles Trapezoid | 1 | 1 (none) | 360° |
| Kite | 1 | 1 (none) | 360° |
| Regular Pentagon | 5 | 5 | 72° |
| Regular Hexagon | 6 | 6 | 60° |
| Regular Octagon | 8 | 8 | 45° |
| Circle | Infinite | Infinite | Any angle |
Types of Line Symmetry
| Type | Description | Examples |
|---|---|---|
| Vertical Line | Line runs up-down, divides left-right | Letters: A, H, I, M, O, T, U, V, W, X, Y |
| Horizontal Line | Line runs left-right, divides top-bottom | Letters: B, C, D, H, I, O, X |
| Diagonal Line | Line at an angle | Square (2 diagonal lines), regular polygons |
| Multiple Lines | More than one line of symmetry | Regular polygons, circles |
Regular Polygons - Quick Reference
| Polygon | Sides (n) | Lines of Symmetry | Rotational Order | Angle of Rotation |
|---|---|---|---|---|
| Equilateral Triangle | 3 | 3 | 3 | 120° |
| Square | 4 | 4 | 4 | 90° |
| Regular Pentagon | 5 | 5 | 5 | 72° |
| Regular Hexagon | 6 | 6 | 6 | 60° |
| Regular Heptagon | 7 | 7 | 7 | ≈51.43° |
| Regular Octagon | 8 | 8 | 8 | 45° |
| Regular Nonagon | 9 | 9 | 9 | 40° |
| Regular Decagon | 10 | 10 | 10 | 36° |
| Regular n-gon | n | n | n | $\frac{360°}{n}$ |
Key Formulas Summary
| Concept | Formula | Use |
|---|---|---|
| Lines of Symmetry (Regular Polygon) | $n$ lines where $n$ = number of sides | Count lines in regular polygons |
| Order of Rotational Symmetry | $\text{Order} = \frac{360°}{\text{Angle}}$ | Find how many times figure matches in 360° |
| Angle of Rotation | $\text{Angle} = \frac{360°}{\text{Order}}$ | Find smallest rotation angle |
| Regular Polygon Rotation | Order = $n$ (number of sides) | Regular n-gon has order n symmetry |
Comparison: Line vs Rotational Symmetry
| Property | Line Symmetry | Rotational Symmetry |
|---|---|---|
| Definition | Reflection across a line | Rotation around a point |
| Transformation | Flip (reflection) | Turn (rotation) |
| Axis/Center | Line of symmetry | Center of rotation |
| Measurement | Number of lines | Order (how many times) |
| Regular n-gon | n lines | Order n |
| Test Method | Fold or use mirror | Rotate and check |
| Rectangle Example | 2 lines | Order 2 (180°) |
Success Tips for Symmetry:
✓ Line symmetry: Figure can be folded so both halves match (mirror image)
✓ Rotational symmetry: Figure looks the same after rotation less than 360°
✓ Regular n-gon has n lines of symmetry AND order n rotational symmetry
✓ Order of rotation = 360° ÷ angle of rotation
✓ Angle of rotation = 360° ÷ order
✓ Circle has INFINITE lines of symmetry and infinite rotational symmetry
✓ Line of symmetry = perpendicular bisector of segments connecting mirror points
✓ Every regular polygon: sides = lines of symmetry = rotational order
✓ To count lines: Look for vertical, horizontal, and diagonal possibilities
✓ Remember: Order 1 = no rotational symmetry (only matches at 360°)
✓ Line symmetry: Figure can be folded so both halves match (mirror image)
✓ Rotational symmetry: Figure looks the same after rotation less than 360°
✓ Regular n-gon has n lines of symmetry AND order n rotational symmetry
✓ Order of rotation = 360° ÷ angle of rotation
✓ Angle of rotation = 360° ÷ order
✓ Circle has INFINITE lines of symmetry and infinite rotational symmetry
✓ Line of symmetry = perpendicular bisector of segments connecting mirror points
✓ Every regular polygon: sides = lines of symmetry = rotational order
✓ To count lines: Look for vertical, horizontal, and diagonal possibilities
✓ Remember: Order 1 = no rotational symmetry (only matches at 360°)