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.

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.

---

Rigid Transformations: Preserving Congruence

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

Translation

A translation is a sliding motion where 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 the horizontal shift and $b$ is the vertical shift
• Vector notation describes the translation as $\langle a, b \rangle$, which is useful for composition problems and proofs
• For example, the vector $\langle 4, -3 \rangle$ shifts every point 4 units right and 3 units down

Reflection

A reflection creates a mirror image across a line of reflection. Each point and its image are equidistant from that line, on opposite sides.

• Over the x-axis: $(x, y) \rightarrow (x, -y)$
• Over the y-axis: $(x, y) \rightarrow (-x, y)$
• Over the line $y = x$: $(x, y) \rightarrow (y, x)$
• Over the line $y = -x$: $(x, y) \rightarrow (-y, -x)$
• Orientation reverses — if vertices were labeled clockwise, the image labels counterclockwise. This is how you distinguish a reflection from a rotation.

Rotation

A rotation is a turning motion around a fixed center point by a specified angle. By convention, positive angles rotate counterclockwise.

• 90° CCW about the origin: $(x, y) \rightarrow (-y, x)$
• 180° about the origin: $(x, y) \rightarrow (-x, -y)$
• 270° CCW (or 90° CW) about the origin: $(x, y) \rightarrow (y, -x)$
• Orientation is preserved — unlike reflection, the clockwise/counterclockwise vertex labeling stays the same

---

Non-Rigid Transformation: Creating Similarity

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

Dilation

A dilation scales a figure from a fixed center of dilation by a scale factor $k$.

• $k > 1$ enlarges the figure, $0 < k < 1$ reduces it, and $k < 0$ reflects through the center and resizes
• Coordinate rule (center at the origin): $(x, y) \rightarrow (kx, ky)$
• Distances multiply by $|k|$ while angle measures stay exactly the same — this is why dilated figures are similar, not congruent
• If the center of dilation is not the origin, say at point $(h, j)$, the rule becomes $(x, y) \rightarrow (k(x - h) + h,\; k(y - j) + j)$

---

Combining Transformations

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

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, just like function composition in algebra.
• A glide reflection is a special composition: a translation followed by a reflection over a line parallel to the translation direction. This is the only rigid transformation (besides the three basic ones) that can't be reduced to a single reflection, rotation, or translation.

Working Through a Composition Step by Step

Here's how to handle a composition problem like "translate $(x, y) \rightarrow (x + 3, y - 2)$, then reflect over the x-axis":

1. Apply the first transformation to get the intermediate image: $(x, y) \rightarrow (x + 3, y - 2)$
2. Apply the second transformation to that result: $(x + 3, y - 2) \rightarrow (x + 3, -(y - 2))$
3. Simplify: the combined rule is $(x, y) \rightarrow (x + 3, -y + 2)$

This coordinate notation lets you verify algebraically that distances and angles are preserved (or scaled, if a dilation is involved).

---

Properties and Applications

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

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

These definitions are central to modern geometry and show up frequently on proofs:

• Congruence: Two figures are congruent if and only if one maps to the other through a sequence of rigid transformations.
• Similarity: Two figures are similar if and only if one maps to the other through a sequence of rigid transformations and a dilation.
• On proofs and justifications, describe the specific transformation sequence rather than just stating figures "look the same." For example: "Reflect $\triangle ABC$ over the y-axis, then translate by $\langle 2, 0 \rangle$ to map it onto $\triangle DEF$."

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 (for a regular $n$-gon, this is $\frac{360°}{n}$).
• Point symmetry is the special case of 180° rotational symmetry — every point has a matching point directly opposite the center, equidistant from it.

---

Advanced: Matrix Representation

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

Transformation Matrices

A 2×2 matrix represents a transformation when multiplied by a coordinate column vector:

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}$

Common transformation matrices (all centered at the origin):

To compose transformations using matrices, multiply the matrices in reverse order of the transformation sequence. If you apply $R$ first and then $T$, compute $T \cdot R$.

---

Quick Reference Table

---

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?