Geometric Transformations

Grade 10 · Math · Study Guide

Study reflections, rotations, translations, and dilations and their effects on coordinates and properties.

A geometric transformation maps every point of a figure to a new position. The original figure is the pre-image; the result is the image. Transformations are classified as rigid (isometric) if they preserve distances and angles, or non-rigid (non-isometric) if they do not.

The four fundamental transformations are: Translation (a slide by a vector (a, b): (x, y) → (x+a, y+b)), Reflection (a flip over a line of symmetry), Rotation (a turn about a centre point by an angle), and Dilation (a scaling from a centre by a factor k: (x, y) → (kx, ky) for centre at origin).

Key coordinate rules: reflection over the x-axis: (x, y) → (x, −y); over the y-axis: (x, y) → (−x, y); over y = x: (x, y) → (y, x). Rotation by 90° counterclockwise about the origin: (x, y) → (−y, x); by 180°: (x, y) → (−x, −y).

Translations, reflections, and rotations are rigid motions (congruence transformations) — they preserve length and angle measure. Dilations preserve angles but change lengths, producing similar figures.

Composite transformations apply two or more transformations in sequence. A glide reflection (translation followed by reflection) is a common composite. The order of composition matters; generally T₂ ∘ T₁ ≠ T₁ ∘ T₂.

Key terms

Rigid Motion (Isometry)
A transformation that preserves all distances and angle measures; includes translations, reflections, and rotations.
Dilation
A transformation that scales a figure by a factor k from a fixed centre; preserves shape (angles) but not size.
Pre-image and Image
The pre-image is the original figure; the image is the figure after transformation.
Line of Reflection
The line that acts as a mirror in a reflection; every point and its image are equidistant from this line.
Centre of Rotation
The fixed point about which a figure is rotated.
Composite Transformation
Two or more transformations applied in succession; the result depends on the order of application.

Flashcards

Translation Rule
(x, y) → (x+a, y+b) moves the figure a units right and b units up.
Reflection over x-axis
(x, y) → (x, −y).
Reflection over y-axis
(x, y) → (−x, y).
Reflection over y = x
(x, y) → (y, x).
Rotation 90° CCW (origin)
(x, y) → (−y, x).
Rotation 180° (origin)
(x, y) → (−x, −y).
Rotation 270° CCW (origin)
(x, y) → (y, −x).
Dilation (origin, factor k)
(x, y) → (kx, ky).

Questions

  1. Point A(3, −2) is reflected over the y-axis. What are the coordinates of A'?

    1. (−3, −2)
    2. (3, 2)
    3. (−3, 2)
    4. (2, −3)
    Show answer

    Answer: (−3, −2)

    Reflection over the y-axis negates the x-coordinate: (3, −2) → (−3, −2).

  2. A rotation of 90° counterclockwise about the origin maps (x, y) to (______, x).

    Show answer

    Answer: -y

    The 90° CCW rotation rule is (x, y) → (−y, x).

  3. A dilation with centre at the origin and scale factor 3 maps point B(2, −1) to:

    1. (6, −3)
    2. (5, 2)
    3. (2/3, −1/3)
    4. (6, 3)
    Show answer

    Answer: (6, −3)

    Dilation by factor 3 from origin: (2×3, −1×3) = (6, −3).

  4. A dilation is a rigid motion because it preserves angle measures.

    Show answer

    Answer: False

    A rigid motion must preserve distances (lengths). A dilation changes lengths by factor k, so it is NOT a rigid motion, even though it preserves angles.

  5. Which transformation maps every point (x, y) to (x + 4, y − 1)?

    1. Translation by vector (4, −1)
    2. Rotation of 4°
    3. Reflection over y = −1
    4. Dilation by factor 4
    Show answer

    Answer: Translation by vector (4, −1)

    The rule (x, y) → (x+4, y−1) adds a constant to each coordinate, which is the definition of a translation by vector (4, −1).

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