6.5 Geometric Transformations & Symmetry

Geometric Transformations & Symmetry

Geometric transformations describe how shapes move, change size, or reorient in a coordinate plane or 2D space. Understanding transformations, congruence, similarity, and symmetry is critical for teaching spatial reasoning and solving coordinate geometry problems on Praxis 5003.

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1. Overview of Geometric Transformations

A transformation is a mathematical operation that maps a original 2D shape (the pre-image) onto a new shape (the image). Standard notation uses prime symbols for the image (e.g., pre-image triangle $ABC \rightarrow$ image triangle $A'B'C'$).

Transformations are categorized into two primary types:

                            ┌───────────────────────────┐
                            │ Geometric Transformations │
                            └─────────────┬─────────────┘
                                          │
                 ┌────────────────────────┴────────────────────────┐
                 │                                                 │
   ┌─────────────┴─────────────┐                     ┌─────────────┴─────────────┐
   │   Rigid Transformations   │                     │ Non-Rigid Transformations │
   │       (Isometries)        │                     │   (Preserves shape only)  │
   │  Preserves Size & Shape   │                     │      E.g., Dilation       │
   └─────────────┬─────────────┘                     └───────────────────────────┘
                 │
      ┌──────────┼──────────┐
      │          │          │
┌─────┴─────┐┌───┴───┐┌─────┴─────┐
│Translation││Reflection│ Rotation│
└───────────┘└───────┘└───────────┘

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2. Rigid Transformations (Isometries)

A rigid transformation (or isometry) is a movement that preserves both distance (side lengths) and angle measures. Under an isometry, the pre-image and image are congruent ($\cong$).

1. Translation (Slide)

A translation shifts every point of a shape by the same distance in a specified direction.

• Coordinate Rule: $(x, y) \rightarrow (x + a, y + b)$, where $a$ is the horizontal shift (right positive, left negative) and $b$ is the vertical shift (up positive, down negative).
• Example: Translating point $P(3, -2)$ by 4 units left and 5 units up gives $P'(3 - 4, -2 + 5) = P'(-1, 3)$.

2. Reflection (Flip)

A reflection flips a shape over a line called the line of reflection. Each point on the image is the same perpendicular distance from the line of reflection as the corresponding point on the pre-image.

• Common Coordinate Rules for Reflections:

3. Rotation (Turn)

A rotation turns a shape around a fixed point called the center of rotation by a specified angle ($\theta$). Unless noted otherwise, positive rotations are counterclockwise (CCW).

• Common Coordinate Rules for Rotations (Center at Origin $(0,0)$):

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3. Non-Rigid Transformations: Dilations

A dilation is a transformation that resizes a shape (enlarging or reducing it) while maintaining its proportional shape. Dilations preserve angle measures, but NOT side lengths. Therefore, dilations produce similar figures ($\sim$), not congruent figures.

• Scale Factor ($k$): The constant ratio of image side length to pre-image side length ($k = \frac{\text{image length}}{\text{pre-image length}}$).
  • If $k > 1$, the dilation is an enlargement.
  • If $0 < k < 1$, the dilation is a reduction.
• Coordinate Rule (Center at Origin): $(x, y) \rightarrow (kx, ky)$.
  • Example: Dilating triangle vertex $A(6, -4)$ by scale factor $k = \frac{1}{2}$ yields $A'(\frac{1}{2} \cdot 6, \frac{1}{2} \cdot -4) = A'(3, -2)$.

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4. Congruence vs. Similarity Summary

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5. Symmetry: Line and Rotational Symmetry

Line Symmetry (Reflectional Symmetry)

A figure has line symmetry if it can be divided by a line (axis of symmetry) into two identical mirror-image halves.

• Square: 4 lines of symmetry (2 diagonal, 1 horizontal, 1 vertical).
• Rectangle: 2 lines of symmetry (horizontal and vertical; diagonals are NOT lines of symmetry!).
• Equilateral Triangle: 3 lines of symmetry.
• Regular $n$-gon: Exactly $n$ lines of symmetry.
• Circle: Infinitely many lines of symmetry (any line passing through the center).

Rotational Symmetry

A figure has rotational symmetry if it can be rotated around a center point by an angle strictly less than $360^\circ$ and coincide exactly with its original position.

• Order of Rotation ($n$): The number of times a shape aligns with itself during a full $360^\circ$ turn.
• Angle of Rotation ($\theta$): The smallest angle through which the shape turns to map onto itself:

$$\text{Angle of Rotation} = \frac{360^\circ}{\text{Order of Rotation } n}$$

Symmetry Properties of Regular Shapes