Composition of Translation and Reflection
Combine two transformations into one: two parallel reflections give a translation of 2d, and two intersecting reflections give a rotation of 2 theta.
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A composition of transformations means applying one transformation after another. The result is a single new transformation — and sometimes two simple moves combine into a familiar one.
Theorem
Order matters
Transformations are applied in sequence, and the order can change the result — a composition is not always commutative.
For example, translating a figure and then reflecting it usually lands it somewhere different from reflecting first and then translating. Always apply the moves in the order given.
Case 1
Two parallel reflections → a translation
Reflecting a figure over one line and then over a second line parallel to it is the same as a single translation. The shift is twice the distance between the two lines, in the direction perpendicular to them.
Two parallel mirrors, distance d apart:
Case 2
Two intersecting reflections → a rotation
Reflecting over two lines that cross is the same as a single rotation about their intersection point. The turn is twice the angle between the two lines.
Two mirrors meeting at angle θ:
Example
Three worked cases
Translation, then reflection
A(2, 1), translate by (3, 2): %20%3D%20(5%2C%5C%203)%20%7D)
.
Then reflect over the x-axis: %20%5Cto%20(5%2C%5C%20-3)%20%7D)
.
⟹ A″ = (5, −3).
Two parallel reflections
Mirrors x = 2 and x = 6, distance 
.
Net shift 
in the +x direction.
⟹ a translation of 8 units along the x-axis.
Two intersecting reflections
Mirrors the x-axis and the line y = x, meeting at angle 45°.
Net turn 
about the origin.
⟹ Q(4, 0) rotates to (0, 4) — a 90° rotation.
Summary
- A composition applies transformations one after another; the order can matter.
- Two reflections over parallel lines equal a translation of
, where d is the distance between them.
- Two reflections over intersecting lines equal a rotation of
about the intersection point.
- Golden rule: transformations are not always commutative — respect the given order.