Sequences of Transformations

One transformation moves a figure once. A sequence of transformations applies several moves in order: the of the first step becomes the of the second, and so on. Use Next step on the figure above to play the sequence — each earlier position stays behind as a faded ghost so you can trace the whole journey.

Does the order matter?

Need a refresher? Negative numbers

Picture the number line: positives to the right of zero, negatives to the left.

Adding moves right: −3 + 5 = 2 (start at −3, take 5 steps right).
Subtracting moves left: −3 − 4 = −7 (start at −3, take 4 steps left).
Subtracting a negative flips to adding: 5 − (−2) = 5 + 2 = 7.

A sign flip turns a number into its mirror image across zero: the opposite of 3 is −3, and the opposite of −3 is 3.

Try it: What is −2 − 6?

Here is a question worth pausing on before you read ahead. Make a prediction, then check it against the arithmetic.

Make a prediction: Take the point (1, 2). You translate by ⟨2, 0⟩ and rotate 90° counter-clockwise about the origin — and a friend does the same two moves in the opposite order. Do you both land on the same point?

No — the two orders usually disagreeYes — the order of transformations never mattersYes, but only because both are rigid motions

Why should swapping the order change anything? The rotation pivots about a fixed point — the origin — so where a figure is when you spin it changes where it lands. Translate first and you spin from a new spot; spin first and you slide a figure that's already turned.

Sequences and congruence

If every step of a sequence is a (a translation, reflection, or rotation), then distances and angles survive every step, so the final image is to the original figure. In fact, that is the definition: two figures are congruent exactly when some sequence of rigid motions carries one onto the other.

Sneak one dilation into the sequence, though, and lengths get scaled — the final image is to the original instead.

Puzzle: get the triangle onto the target

Build a sequence that carries $\triangle ABC$ exactly onto the dashed target. Two moves are enough — but the order is up to you to discover.

Build a sequence that carries the figure onto the target.

Tap a move to add it to your sequence

Your sequence (0 of 3)

Tap a move above to begin building your sequence →

Sequences of Transformations

Does the order matter?

Sequences and congruence

Puzzle: get the triangle onto the target

Practice