Transformations in the Coordinate Plane

Why This Matters

Transformations are the foundation for understanding two of geometry's biggest ideas: congruence and similarity. When you move, flip, rotate, or resize a figure, you're not just manipulating shapes—you're proving relationships between them. Every time an exam asks you to show two triangles are congruent or explain why figures are similar, transformations give you the tools to answer with precision. You'll see these concepts tested through coordinate proofs, mapping rules, and composition problems.

The key insight is that transformations fall into two categories: those that preserve everything about a shape (rigid transformations) and those that change size while keeping proportions (dilations). This distinction determines whether you're dealing with congruent or similar figures. Don't just memorize the formulas—know which transformation proves what and how the coordinate rules reflect the geometric action.

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Rigid Transformations: Preserving Congruence

Rigid transformations—also called isometries—keep distance and angle measures intact. The original figure and its image are always congruent, meaning identical in size and shape.

Translation

• Sliding motion—every point moves the same distance in the same direction, with no turning or flipping
• Coordinate rule: $(x, y) \rightarrow (x + a, y + b)$ where $a$ is horizontal shift and $b$ is vertical shift
• Vector notation describes the translation as $\langle a, b \rangle$, useful for composition problems and proofs

Reflection

• Mirror image across a line of reflection—each point and its image are equidistant from the line
• Common rules: reflection over x-axis: $(x, y) \rightarrow (x, -y)$; over y-axis: $(x, y) \rightarrow (-x, y)$; over $y = x$: $(x, y) \rightarrow (y, x)$
• Orientation reverses—if vertices were labeled clockwise, the image labels counterclockwise (this is how you distinguish reflection from rotation)

Rotation

• Turning motion around a center point by a specified angle; positive angles rotate counterclockwise
• Rules about the origin: 90°: $(x, y) \rightarrow (-y, x)$; 180°: $(x, y) \rightarrow (-x, -y)$; 270°: $(x, y) \rightarrow (y, -x)$
• Orientation preserved—unlike reflection, the clockwise/counterclockwise labeling stays the same

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Non-Rigid Transformation: Creating Similarity

Dilation is the only transformation that changes size. It produces similar figures—same shape, proportional dimensions, but different measurements.

Dilation

• Scale factor $k$ determines the size change: $k > 1$ enlarges, $0 < k < 1$ reduces, $k < 0$ reflects and resizes
• Coordinate rule from origin: $(x, y) \rightarrow (kx, ky)$—multiply both coordinates by the scale factor
• Distances multiply by $|k|$ while angle measures stay exactly the same—this is why dilated figures are similar, not congruent

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Combining Transformations

When you perform multiple transformations in sequence, the result depends on both which transformations and what order.

Composition of Transformations

• Order matters—reflecting then translating often gives a different result than translating then reflecting
• Notation: $(T \circ R)(x, y)$ means apply $R$ first, then $T$; read right to left like function composition
• Glide reflection is a special composition: translation followed by reflection over a line parallel to the translation direction

Coordinate Notation for Transformations

• Mapping rule format $(x, y) \rightarrow (x', y')$ precisely describes how every point transforms
• Combining rules: for translation $(x, y) \rightarrow (x + 3, y - 2)$ followed by reflection over x-axis, the composition is $(x, y) \rightarrow (x + 3, -(y - 2))$
• Essential for proofs—coordinate notation lets you verify that distances and angles are preserved algebraically

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Properties and Applications

Understanding the underlying properties helps you recognize transformations and apply them strategically.

Properties of Rigid Transformations

• Distance preservation—the length of any segment equals the length of its image ($AB = A'B'$)
• Angle preservation—corresponding angles remain congruent, so shape is maintained exactly
• Betweenness and collinearity preserved—if point $B$ is between $A$ and $C$, then $B'$ is between $A'$ and $C'$

Congruence and Similarity Through Transformations

• Congruence definition: two figures are congruent if one maps to the other through a sequence of rigid transformations
• Similarity definition: two figures are similar if one maps to the other through rigid transformations and a dilation
• Proving relationships—describe the specific transformation sequence rather than just stating figures "look the same"

Symmetry in the Coordinate Plane

• Line symmetry exists when a figure maps onto itself via reflection—the reflection line is the axis of symmetry
• Rotational symmetry exists when a figure maps onto itself via rotation of less than 360°; the smallest such angle is the angle of symmetry
• Point symmetry is the special case of 180° rotational symmetry—every point has a matching point equidistant from the center

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Advanced: Matrix Representation

For students ready to connect geometry to linear algebra, matrices provide a powerful computational tool.

Transformation Matrices

• 2×2 matrices represent transformations when multiplied by coordinate vectors: $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}$
• Common matrices: 90° rotation: $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$; reflection over x-axis: $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$; dilation by $k$: $\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$
• Matrix multiplication handles composition—multiply matrices in reverse order of transformation sequence

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Quick Reference Table

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Self-Check Questions

1. Which two transformations both preserve congruence but differ in how they affect orientation? Explain the difference.

2. A figure is mapped using $(x, y) \rightarrow (2x, 2y)$ then $(x, y) \rightarrow (x + 3, y - 1)$. Is the final image congruent or similar to the original? Justify your answer.

3. Compare and contrast line symmetry and rotational symmetry. Give an example of a figure that has one but not the other.

4. If two triangles are similar but not congruent, what type(s) of transformations must be included in the sequence mapping one to the other?

5. The composition of two reflections over parallel lines produces what single transformation? What if the lines intersect?