Introduction
Symmetry is a property of a figure where it can be divided into two halves that are mirror images of each other. When folded along a specific line, called the line of symmetry, the two halves coincide exactly.
Line of Symmetry: A line that divides a figure into two identical parts such that one half is the mirror image of the other.
Reflection: The phenomenon where an image is formed as a mirror image of an object, preserving lengths and angles.
Rotational Symmetry: A figure has rotational symmetry if it looks the same after being rotated about a fixed point (called the center of rotation) by a certain angle less than 360°.
Formula Derivation
Reflection symmetry involves a line such that for every point \( P \) on one side, there exists a point \( P' \) on the other side such that the line is the perpendicular bisector of segment \( PP' \).
Rotational symmetry is characterized by the angle \( \theta = \frac{360^\circ}{n} \), where \( n \) is the order of rotational symmetry (number of times the figure maps onto itself during a full rotation).
Worked Illustration
Consider a square. It has 4 lines of symmetry (vertical, horizontal, and two diagonals) and rotational symmetry of order 4 with angle 90°.
Solved Example
Example: Find the order and angle of rotational symmetry of a square.
Solution:
The square maps onto itself 4 times during a full 360° rotation.
Order \( n = 4 \)
Angle of rotational symmetry \( \theta = \frac{360^\circ}{4} = 90^\circ \)
Practice Set
- Level 1 – Easy: Identify the lines of symmetry in the letters A, H, and T.
- Level 2 – Moderate: Determine the order of rotational symmetry of an equilateral triangle.
- Level 3 – Challenging: Prove that a regular pentagon has 5 lines of symmetry and rotational symmetry of order 5.
Answer Key
- Level 1: A has 1 vertical line, H has 2 lines (vertical and horizontal), T has 1 vertical line.
- Level 2: Equilateral triangle has rotational symmetry order 3, angle 120°.
- Level 3: Each line of symmetry passes through a vertex and the midpoint of the opposite side; rotational symmetry order is 5 with angle 72°.
Quick Reference
| Concept | Definition | Formula/Property |
|---|---|---|
| Line of Symmetry | Line dividing figure into mirror halves | Perpendicular bisector of corresponding points |
| Rotational Symmetry | Figure maps onto itself on rotation | Angle \( = \frac{360^\circ}{n} \), Order \( n \) |
Glossary
- Symmetry: Balanced and proportionate similarity between halves of a figure.
- Line of Symmetry: A line dividing a figure into two identical mirror-image parts.
- Reflection: Mirror image formation preserving size and shape.
- Rotational Symmetry: Property of a figure to look the same after rotation by certain angles.
- Order of Rotational Symmetry: Number of times a figure coincides with itself during a full rotation.
- Angle of Rotational Symmetry: Smallest angle of rotation for which the figure maps onto itself.
Symmetry
Symmetry in a figure means parts of the figure are repeated in a definite pattern such that the figure looks balanced and identical on folding or rotation.
Formula Derivation
For reflection symmetry, if \( L \) is the line of symmetry, then for any point \( P(x,y) \), its mirror image \( P'(x',y') \) satisfies:
\[ d(P,L) = d(P',L) \quad \text{and} \quad L \text{ is the perpendicular bisector of } PP' \]
Worked Illustration
The Taj Mahal is an example of architectural symmetry, where the left and right halves are mirror images about a central vertical line.
Solved Example
Example: Show that the butterfly figure has a vertical line of symmetry.
Solution: Folding the butterfly along the vertical line divides it into two halves that coincide exactly, confirming vertical symmetry.
Practice Set
- Level 1: Identify symmetry in natural objects like leaves and flowers.
- Level 2: Draw a figure with exactly two lines of symmetry.
- Level 3: Prove that the letter "H" has two lines of symmetry.
Answer Key
- Level 1: Many leaves have bilateral symmetry.
- Level 2: A rectangle has two lines of symmetry (vertical and horizontal).
- Level 3: "H" is symmetric about vertical and horizontal lines through its center.
Quick Reference
| Type | Example | Lines of Symmetry |
|---|---|---|
| Vertical | Butterfly | 1 |
| Horizontal | Arrow | 1 |
| Multiple | Square | 4 |
Glossary
- Symmetry: Balanced repetition of parts in a figure.
- Vertical Line of Symmetry: Line dividing figure into left and right mirror halves.
- Horizontal Line of Symmetry: Line dividing figure into top and bottom mirror halves.
- Diagonal Line of Symmetry: Line dividing figure diagonally into mirror halves.
Line of Symmetry
The line of symmetry divides a figure into two identical parts that overlap exactly when folded along the line.
Formula Derivation
For a line of symmetry \( L \), any point \( P \) on one side has a corresponding point \( P' \) on the other side such that \( L \) is the perpendicular bisector of segment \( PP' \).
Worked Illustration
Letters like A, H, and M have vertical lines of symmetry, while letters like B and C have horizontal lines of symmetry.
Solved Example
Example: Identify the lines of symmetry in the letter "H".
Solution: The letter "H" has two lines of symmetry: one vertical and one horizontal, both passing through its center.
Practice Set
- Level 1: Draw the line of symmetry for the letter "A".
- Level 2: Find the number of lines of symmetry in an equilateral triangle.
- Level 3: Prove that a regular pentagon has five lines of symmetry.
Answer Key
- Level 1: One vertical line through the center of "A".
- Level 2: Three lines of symmetry, each passing through a vertex and the midpoint of the opposite side.
- Level 3: Each line passes through a vertex and the midpoint of the opposite side, totaling five lines.
Quick Reference
| Shape | Lines of Symmetry |
|---|---|
| Equilateral Triangle | 3 |
| Square | 4 |
| Regular Pentagon | 5 |
Glossary
- Line of Symmetry: Line dividing a figure into two identical mirror-image parts.
- Vertical Line of Symmetry: Line dividing figure into left and right halves.
- Horizontal Line of Symmetry: Line dividing figure into top and bottom halves.
- Diagonal Line of Symmetry: Line dividing figure diagonally into mirror halves.
Reflection
Reflection is the process where a figure is flipped over a line (line of symmetry) to produce a mirror image.
Formula Derivation
If the line of symmetry is the y-axis, the reflection of a point \( (x,y) \) is \( (-x,y) \). For a general line, reflection formulas depend on the line's equation.
Worked Illustration
The letter "M" reflected in a plane mirror appears reversed left to right.
Solved Example
Example: Find the reflection of point \( (3,4) \) about the y-axis.
Solution: Reflection about y-axis changes \( x \) to \( -x \), so the image is \( (-3,4) \).
Practice Set
- Level 1: Reflect the point \( (2,5) \) about the x-axis.
- Level 2: Find the reflection of \( (1,-3) \) about the line \( y = x \).
- Level 3: Prove that reflection preserves distances and angles.
Answer Key
- Level 1: Reflection about x-axis gives \( (2,-5) \).
- Level 2: Reflection about \( y = x \) swaps coordinates: \( (-3,1) \).
- Level 3: Reflection is an isometry; it preserves distances and angles by definition.
Quick Reference
| Reflection Line | Image of Point \( (x,y) \) |
|---|---|
| y-axis | \( (-x,y) \) |
| x-axis | \( (x,-y) \) |
| y = x | \( (y,x) \) |
Glossary
- Reflection: Flipping a figure over a line to produce a mirror image.
- Plane Mirror: A flat reflective surface producing virtual images.
- Isometry: A transformation preserving distances and angles.
Rotational Symmetry
A figure has rotational symmetry if it can be rotated about a fixed point (center of rotation) by an angle less than 360° and still look exactly the same.
Formula Derivation
The order of rotational symmetry \( n \) is the number of times the figure coincides with itself during a full 360° rotation.
The angle of rotational symmetry \( \theta \) is given by:
\[ \theta = \frac{360^\circ}{n} \]
Worked Illustration
A square has rotational symmetry of order 4 with angles 90°, 180°, 270°, and 360°.
Solved Example
Example: Find the order and angle of rotational symmetry of a pinwheel with 4 blades.
Solution:
The pinwheel looks the same after rotations of 90°, 180°, 270°, and 360°.
Order \( n = 4 \)
Angle \( \theta = \frac{360^\circ}{4} = 90^\circ \)
Practice Set
- Level 1: Identify the order of rotational symmetry of an equilateral triangle.
- Level 2: Find the angle of rotational symmetry of a regular hexagon.
- Level 3: Prove that a circle has infinite order of rotational symmetry.
Answer Key
- Level 1: Order 3, angle 120°.
- Level 2: Order 6, angle 60°.
- Level 3: A circle coincides with itself at every angle of rotation, so infinite order.
Quick Reference
| Shape | Order of Rotational Symmetry | Angle of Rotational Symmetry |
|---|---|---|
| Equilateral Triangle | 3 | 120° |
| Square | 4 | 90° |
| Regular Hexagon | 6 | 60° |
| Circle | Infinite | Any angle |
Glossary
- Rotational Symmetry: Property of a figure to look the same after rotation about a point.
- Order of Rotational Symmetry: Number of times a figure coincides with itself in one full rotation.
- Angle of Rotational Symmetry: Smallest angle for which the figure maps onto itself.
- Center of Rotation: Fixed point about which rotation occurs.