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Geometry Transformations, congruence, and proof

Compositions of transformations

20 practice questions 2 video lessons Theory + worked examples

Compositions of Transformations

Common Core Geometry • Standard G-CO.5 • Transformations, Congruence & Proof

Compositions of Transformations is a topic in Transformations, Congruence & Proof in the Common Core State Standards. It is aligned to Standard G-CO.5, which requires students to draw and specify the sequence of transformations that will carry one figure onto another.

A composition of transformations applies one transformation after another; a composition of rigid motions is itself a single rigid motion.

Common Core Geometry › Transformations, Congruence & Proof › Compositions of Transformations  —  Standard G-CO.5

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Practice questions

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  • Composition of Transformations: Examples (Geometry Concepts) Watch
  • Compositions of Transformations - Combining Multiple Transformations Watch
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Theory

A composition of transformations applies one transformation, then another, to the result. The order is written right to left but performed in sequence.

Key facts about combining reflections:

  • Two reflections over parallel lines \(=\) a translation (twice the distance between the lines).
  • Two reflections over intersecting lines \(=\) a rotation (twice the angle between the lines).
  • A glide reflection is a translation along a line combined with a reflection over that line.
A composition of rigid motions is still a rigid motion, so the final image is congruent to the original.
A composition of transformations A composition applies one transformation after another; here a translation of the triangle followed by a reflection over a mirror line. 1. translate 2. reflect mirror line
A composition: translate the triangle, then reflect it.
Two reflections combine Reflecting over two parallel lines is the same as a single translation; over two intersecting lines it is a rotation. line 1 line 2 two reflections over parallel lines = translation
Two reflections over parallel lines equal one translation.

Reflection compositions:

\[\text{parallel lines }d\text{ apart}\ \Rightarrow\ \text{translation of }2d\]
\[\text{lines meeting at }\theta\ \Rightarrow\ \text{rotation of }2\theta\]
two parallel reflections give a translation of twice the distance; two intersecting reflections give a rotation of twice the angle
Order can matter. Unless the transformations commute, swapping them may change the image.

How to perform a composition

  1. Apply the first transformation to every point.
  2. Apply the second to the resulting image.
  3. Describe the single equivalent transformation when one exists (translation, rotation, or glide reflection).
Example 1 — Compose two rules
Apply to \(A(2,3)\): first \((x,y)\to(x+1,y)\), then reflect over the \(x\)-axis.
Solution

Do the transformations in order.

\(\text{translate}\)\(:\)\((2,3)\to(3,3)\)
\(\text{reflect } x\)\(:\)\((3,3)\to(3,-3)\)

Final image: \((3,-3)\).

the final image is 3 comma negative 3
Example 2 — Order matters
Does reversing the order in Example 1 give the same result for \(A(2,3)\)?
Solution

Reflect first, then translate.

\(\text{reflect } x\)\(:\)\((2,3)\to(2,-3)\)
\(\text{translate}\)\(:\)\((2,-3)\to(3,-3)\)

Same point here, but in general the order of a composition can change the result.

here the result matches, but order can matter in general
Example 3 — Two parallel reflections
What single transformation equals a reflection over \(x=1\) followed by a reflection over \(x=4\)?
Solution

Two reflections over parallel lines give a translation of twice the distance between them, in the direction perpendicular to the lines.

\(\text{shift}\)\(=\)\(2(4-1)=6\ \text{units right}\)
the result is a translation of 6 units right
Example 4 — A glide reflection
What is a glide reflection?
Solution

A glide reflection is the composition of a translation along a line and a reflection over that same line — the motion that produces a line of footprints.

a glide reflection is a translation followed by a reflection over the same line

Common pitfalls

Do the transformations in the right order. The result can differ when the order is swapped.
Two parallel reflections make a translation, not a bigger reflection. The shift is twice the gap between the lines.
A glide reflection is one motion, not two separate steps to report.

Frequently asked questions

What is a composition of transformations?

Applying one transformation and then another to the result, producing a single overall motion.

What do two reflections over parallel lines produce?

A translation, of twice the distance between the two lines, perpendicular to them.

What is a glide reflection?

A translation along a line combined with a reflection over that same line — the motion behind a trail of footprints.

Does the order of a composition matter?

Often yes. Unless the transformations commute, performing them in a different order can give a different image.

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