ACT® · Math: Geometry — Transformations, Proof, and Constructions
Composing Two Transformations
Composing two transformations means doing one move, then applying the second move to the result, not to the original figure. Order matters unless the moves are special pairs like two translations.
When a problem stacks two transformations, work inside out: apply the first one to every point, get a new figure, then apply the second one to that new figure. The final coordinates depend on getting this order right, because the second transformation only ever sees the output of the first, never the original points.
Some pairs of transformations combine into a single simpler move, and that is where the real insight lives. Two reflections across parallel lines always equal one translation, perpendicular to the mirrors, by twice the distance between them. Two reflections across perpendicular lines, like the x-axis then the y-axis, always equal a 180 degree rotation about their intersection point, because flipping twice in perpendicular directions sends (x, y) to (-x, y) to (-x, -y), which is the same as rotating 180 degrees about the origin. This is why every rigid motion in the plane, no matter how complicated it looks, reduces to just one of four things: a translation, a rotation, a reflection, or a glide reflection (a reflection followed by a slide along the mirror line).
Order matters whenever rotation is in the mix, because rotating turns everything about the origin, including a translation vector from an earlier step. Translate first then rotate, and the slide direction gets rotated along with the figure. Rotate first then translate, and the slide direction stays fixed. Two translations, by contrast, always commute, since adding vectors in either order gives the same sum. This sits right next to single-transformation work: composing is just chaining those same rules twice and tracking what the first move hands off to the second.
Key terms
- Composition of transformations
- Applying one transformation to a figure, then applying a second transformation to that resulting image.
- Glide reflection
- A reflection across a line followed by a translation parallel to that same line.
- Commute
- Two operations commute if doing them in either order gives the same final result.
- Rigid motion
- A transformation that preserves distances and angles, limited to translations, rotations, reflections, and glide reflections.
Watch it explained
- How it worksHow does a flip across the x-axis followed by a flip across the y-axis produce exactly a 180 degree turn?
- Worked exampleFind where (2, 5) ends up after a reflection across the x-axis followed by a translation 3 left and 4 up.
- Worked exampleWalk me through reflecting (1, 1) across y = x and then rotating that result 90 degrees about the origin.
- Common mistakeWhy doesn't swapping the order of a rotation and a translation always land the figure in the same place?
- Ties togetherHow does composing transformations explain why every rigid motion is a slide, a turn, a flip, or a flip plus a slide?
Check yourself
End of topic.