You can follow one transformation with another. The result, called a composition, is again a transformation: $P \to A(P) \to B(A(P))$. Composing the rigid motions from Precise Definitions of Rigid Motions always gives a rigid motion, since each step keeps distances.
Worked example: Order matters
$P = (2, 1)$. (a) Translate by $\langle 3, 0 \rangle$, then reflect over the $y$-axis. (b) Reflect first, then translate.
- (a) $(2, 1) \to (5, 1) \to (-5, 1)$.
- (b) $(2, 1) \to (-2, 1) \to (1, 1)$.
- The results differ, so these two transformations do not commute.
Answer: (a) $(-5, 1)$ (b) $(1, 1)$
Why it works
Two reflections make something new. Reflect over a line $\ell$, then over a parallel line $m$ a distance $d$ away: a point $x$ units from $\ell$ ends up shifted $2d$ in the direction from $\ell$ to $m$, whatever $x$ is, so the composition is a translation by $2d$. If the two mirror lines cross at $C$ with angle $\theta$ between them, the composition is a rotation about $C$ by $2\theta$. Every rigid motion can be built from at most three reflections.
Worked example: Two reflections over parallel lines
Reflect $P(0, 2)$ over $x = 1$, then over $x = 4$. What single transformation does the same thing?
- First: $P$ is 1 unit left of $x = 1$, so it goes to $(2, 2)$.
- Second: $(2, 2)$ is 2 units left of $x = 4$, so it goes to $(6, 2)$.
- The lines are $3$ units apart, and $P$ moved $6 = 2 \cdot 3$ units right.
Answer: A translation by $\langle 6, 0 \rangle$.
Worked example: Finding a sequence
$\triangle ABC$ has $A(1, 1)$, $B(3, 1)$, $C(1, 2)$. $\triangle DEF$ has $D(-1, -3)$, $E(-3, -3)$, $F(-1, -4)$. Find rigid motions that map $\triangle ABC$ onto $\triangle DEF$ in order.
- Compare shapes: $AB$ points right, but $DE$ points left; $AC$ points up, but $DF$ points down. A $180^\circ$ rotation about the origin flips both: $(x, y) \to (-x, -y)$.
- Rotation: $A \to (-1, -1)$, $B \to (-3, -1)$, $C \to (-1, -2)$.
- Each image is 2 units above its target, so translate by $\langle 0, -2 \rangle$.
Answer: Rotate $180^\circ$ about the origin, then translate by $\langle 0, -2 \rangle$.
Key idea
A composition does one transformation, then the next. The order can change the result. Two reflections in parallel lines make a translation (twice the gap); in crossing lines, a rotation (twice the angle).
Common mistake
Assuming the order never matters. Sometimes it does not (two translations, or two rotations about the same center), but often it does, as in the first example. Check by sending one point through both orders.