# 4.3: Algebra of Transformations

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## 4.3.1Explore

##### Definition: Translation

An transformation T is a translation if and only if there exists a non-zero constant vector v such that T(P) - P = v for all points P.

##### Definition: Rotation

An transformation T is a rotation if and only if there exists a fixed point C and constant angle α such that m∠PCT(P) = α and llCPll = llCT(P)ll for all points P.

##### Definition: Reflection

An transformation T is a reflection if and only if there exists a fixed line ℓ such that the line perpendicular to ℓ through P contains T(P) and the distances from P and T(P) to the ℓ are equal.

##### Theorem

Translations, rotations, and reflections are isometries.

Draw an arbitrary triangle △ABC. Draw the result △A'B'C' of some translation. Draw the result △A''B''C'' of some translation applied to △A'B'C' Determine which type of isometry would transform △ABC to △A''B''C''.

Complete the following table of composition of isometries.

 Translate Reflect Rotate Translate Reflect Rotate

## 4.3.2Prove

How many isometry types are there?

How many isometry types are needed to generate all isometry types?

How many isometries are needed to generate all isometries?

This page titled 4.3: Algebra of Transformations is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Mark A. Fitch via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.