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Geometric Transformations
Translation, Reflection, Rotation, and Dilation
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Geometric transformations move, flip, turn, or resize figures in the coordinate plane. The four rigid motions - translation, reflection, and rotation - preserve shape and size (congruence). Dilation changes size but preserves shape (similarity). Together, they form the language of symmetry and are fundamental in art, architecture, computer graphics, and crystallography.
Coordinate rules make transformations precise: a translation by (h, k) maps (x, y) to (x+h, y+k). A reflection over the y-axis maps (x, y) to (-x, y). A 90° counterclockwise rotation about the origin maps (x, y) to (-y, x). A dilation by factor k centered at the origin maps (x, y) to (kx, ky). Memorizing and applying these rules is the core skill.
Key Facts
- Translation by (h, k): (x, y) → (x+h, y+k)
- Reflection over x-axis: (x, y) → (x, -y)
- Reflection over y-axis: (x, y) → (-x, y)
- Rotation 90° CCW: (x, y) → (-y, x); 180°: (x, y) → (-x, -y); 270° CCW: (x, y) → (y, -x)
- Dilation by factor k about origin: (x, y) → (kx, ky)
- Rigid motions preserve congruence; dilations produce similar (not congruent) figures.
Vocabulary
- Translation
- A slide that moves every point the same distance in the same direction.
- Reflection
- A flip over a line (the line of reflection); produces a mirror image.
- Rotation
- A turn about a fixed point (the center of rotation) by a given angle and direction.
- Dilation
- A scaling transformation that enlarges or reduces a figure by a scale factor k relative to a center point.
- Image
- The resulting figure after a transformation is applied to the pre-image.
Common Mistakes to Avoid
- Reflecting over the wrong axis. Reflection over the x-axis changes the sign of y; over the y-axis changes the sign of x.
- Applying dilation from the wrong center. If the center is not the origin, you must translate to the origin first, dilate, then translate back.
- Confusing 90° clockwise and counterclockwise rotations. CCW 90°: (x, y) → (-y, x). CW 90°: (x, y) → (y, -x).
- Assuming congruence after a dilation. Dilation changes size, so the figures are similar, not congruent (unless the scale factor is 1 or -1).
Practice Questions
- 1 Triangle ABC has vertices A(1,2), B(3,4), C(2,5). After a translation of (-2, 3), what are the new coordinates?
- 2 Reflect the point (4, -3) over the y-axis, then rotate the result 90° counterclockwise about the origin.
- 3 A figure is dilated by a factor of 2.5 about the origin. If one vertex was at (4, -2), where is its image?