Question

Determining Congruence
Do you think a sequence of translations across the x- or y-axis and/or reflections on a figure could result in the same image as a 90-degree clockwise rotation? Explain why or why not.

Answer 1

No. A 90° rotation cannot be made from only translations and reflections across the x- and y-axes.

Reason (simple algebraic invariant). Reflection across the x-axis, reflection across the y-axis, and any translation have the forms
- Rx(x,y) = (x, −y)
- Ry(x,y) = (−x, y)
- T(a,b)(x,y) = (x + a, y + b).

Any composition of these maps has the form
(x,y) ↦ (±x + a, ±y + b),
so each output coordinate depends only on the corresponding input coordinate (x or y), possibly with a sign change and a constant shift. In particular the coordinates are never swapped.

A 90° clockwise rotation about the origin is
(x,y) ↦ (y, −x),
which swaps x and y. Because the generators above never swap coordinates, no sequence of those reflections and translations can produce the 90° rotation.

(Another viewpoint: reflecting across two lines that meet at angle θ gives a rotation by 2θ. To get 90° you would need two lines meeting at 45°, not the x- and y-axes.)