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A rigid motion, or **isometry**, is a transformation that preserves the distance of the points of the preimage. **congruence transformations** because the preimage and its image under a rigid motion are congruent figures. Some examples of rigid motions are translations, reflections, and rotations.

$AB=A_{′}B_{′} $

On the diagram, the logo on the left is the preimage, while the logo on the right is the image. Furthermore, the logo on the right is the result of a rigid motion as the distances between all the points are preserved.
Rigid motions are also called A rigid motion preserves the side lengths and angle measures of a polygon. As a result, a rigid motion maintains the exact size and shape of a figure. Still, a rigid motion can affect the position and orientation of the figure.

- A rigid motion preserves the side lengths of a polygon because, by definition, the distance between the vertices do not change.
- It is accepted without a proof that rigid motions also preserve angle measures.

A translation is a transformation that moves every point of a figure the same distance in the same direction. More precisely, a translation along a vector $v$ maps every point $A$ in the plane onto its image $A_{′}$ such that the following statements hold true.

- The length of $AA_{′}$ and the magnitude of $v$ are equal.
- Segment $AA_{′}$ and $v$ are parallel.
- Vectors $AA_{′}$ and $v$ have the same direction.

Since translations preserve side lengths and angle measures, they are rigid motions. Additionally, translations map lines onto parallel lines.

A reflection is a transformation in which every point of a figure is reflected in a line. The line in which the points are reflected is called the line of reflection and acts like a mirror.

More precisely, a reflection across a line $ℓ$ maps every point $A$ in the plane into its image $A_{′}$ such that one of the following statements is satisfied.

- If $A$ is on the line $ℓ,$ then $A$ and $A_{′}$ are the same point.
- If $A$ is not on the line $ℓ,$ then $ℓ$ is the perpendicular bisector of $AA_{′}.$

Like rotations and translations, reflections are also rigid motions because they preserve the side lengths and angle measures. However, reflections change the orientation of the preimage.

A rotation is a transformation in which a figure is turned about a fixed point $P$ by a certain angle measure $α_{∘}.$ The point $P$ is called the center of rotation. Rotations map every point $A$ in the plane to its image $A_{′}$ such that one of the following statements is satisfied.

- If $A$ is the center of rotation, then $A$ and $A_{′}$ are the same point.
- If $A$ is not the center of rotation, then $A$ and $A_{′}$ are the same distance from $P,$ and $∠APA_{′}$ has a measure of $α_{∘}.$

Usually, rotations are performed counterclockwise unless otherwise stated.

A glide reflection is a transformation that combines a translation and a reflection across a line parallel to the translation vector. Since a glide reflection is a composition of rigid motions, it is also a rigid motion.

Thanks to the fact that the line of reflection and the translation vector are parallel, a glide reflection could instead be a reflection followed by a translation. That is, the image does not depend on the order of the transformations.