Introduces rigid transformations (translations, reflections, rotations) and non-rigid transformations (dilations) on the coordinate plane. Students describe transformations using precise language and coordinates, verify that rigid transformations preserve size and shape, and explore how dilations change size while preserving shape.
Middle School Geometry & Data • 6-8
A transformation moves or changes a figure on the coordinate plane. Some transformations keep the figure's size and shape (rigid), while one type changes size. Mastering transformations means you can predict exactly where every point ends up using simple coordinate rules.
These three transformations produce a figure that is congruent to the original:
Every point moves the same distance in the same direction. Described by a vector (a, b):
Example: Translate by (3, −2) means move right 3, down 2.
The figure is flipped over a line of reflection. Common reflections:
| Reflect over | Rule |
|---|---|
| x-axis | (x, y) → (x, −y) |
| y-axis | (x, y) → (−x, y) |
| Line y = x | (x, y) → (y, x) |
The figure turns around a center point (usually the origin). Positive angles go counterclockwise:
| Rotation | Rule (about origin) |
|---|---|
| 90° counterclockwise | (x, y) → (−y, x) |
| 180° | (x, y) → (−x, −y) |
| 270° counterclockwise (= 90° clockwise) | (x, y) → (y, −x) |
A dilation enlarges or shrinks a figure from a center point by a scale factor k. The shape stays the same (similar), but side lengths multiply by k:
Translations, reflections, and rotations are rigid—the image is congruent to the original. Dilations are non-rigid—the image is similar but not congruent (unless k = 1).
Translate triangle with vertices A(1, 4), B(3, 1), C(5, 4) by the vector (−4, +2).
Rotate point P(3, −2) 90° counterclockwise about the origin.
Dilate triangle with vertices A(2, 4), B(6, 2), C(4, 8) by scale factor k = 1/2, center at origin.
Each side is half its original length. The image is similar but smaller.
When rotating, students often swap the sign changes. Write out the rule before substituting. For 90° CCW: (x, y) → (−y, x). The y-value gets negated and goes first, the x-value goes second.
1. Reflect (4, −7) over the x-axis.
(x, y) → (x, −y) → (4, 7)
2. Translate (−2, 5) by vector (6, −3).
(−2 + 6, 5 + (−3)) = (4, 2)
3. Rotate (1, 5) by 180° about the origin.
(x, y) → (−x, −y) → (−1, −5)
4. A square has vertices at (0,0), (4,0), (4,4), (0,4). Dilate by k = 3 from the origin. What are the new vertices?
(0,0), (12,0), (12,12), (0,12). Each coordinate is multiplied by 3. Side length goes from 4 to 12.
5. Point Q(5, 2) is reflected over the y-axis, then translated by (1, 3). What are the final coordinates?
Reflect over y-axis: (5, 2) → (−5, 2). Translate: (−5 + 1, 2 + 3) = (−4, 5).