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.
Understanding Geometric Transformations
A transformation can be understood by tracking a few important features of a figure. Start with its vertices, since straight sides and angles follow from those points. For a rigid motion, the distance between every pair of vertices stays unchanged.
This means side lengths, angle measures, perimeter, and area remain the same. The order of the vertices matters too. A translation and a rotation keep the figure's orientation.
If the vertices go around clockwise before the move, they still go clockwise afterward. A reflection reverses orientation, which is why a reflected letter or shape can look like a mirror image rather than a simple turn.
The center or line used for a transformation is not a minor detail. A rotation is defined by both an angle and a center. Points farther from the center travel along larger circular paths, yet each point stays the same distance from that center.
When the center is not the origin, students can shift the figure so the center acts like the origin, rotate it, then shift it back. Reflections work in a similar way.
Every point and its image lie the same perpendicular distance from the mirror line. The mirror line cuts the segment joining them exactly in half.
Dilations need careful attention because the scale factor describes distance from the center, not a fixed amount added to each coordinate. A factor greater than one sends points farther away. A positive factor between zero and one pulls points closer.
Corresponding side lengths are multiplied by the factor, while area is multiplied by the factor times itself. For example, doubling every length makes an area four times as large.
A negative scale factor places the image on the opposite side of the center. This can feel like a dilation combined with a half turn, even though the size rule still controls the result.
Many real tasks use several transformations in sequence. A pattern on wallpaper may repeat through translations, while a logo may use reflections and rotations to create balance. In computer graphics, an object is often resized, turned, then placed on a screen.
The order matters. Turning a figure and then shifting it usually gives a different result from shifting it and then turning it. On graph problems, label the original points clearly and make a table for their images.
Check one distance, one angle, and the location of the center or mirror line. These checks catch sign errors before they spread through the whole figure.
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?