A rigid motion, or isometry, is a transformation that preserves the distance of the points of the preimage. 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 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.

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 $\vec{v}$ maps every point $A$ in the plane onto its image $A^{\prime }$ such that the following statements hold true.

• The length of $\overline{A{A}^{\prime }}$ and the magnitude of $\vec{v}$ are equal.
• Segment $\overline{A{A}^{\prime }}$ and $\vec{v}$ are parallel.
• Vectors $\overrightarrow{A{A}^{\prime }}$ and $\vec{v}$ have the same direction.

Notice that these three properties imply that the quadrilateral formed by $A,$ $A^{\prime },$ the tip of $\vec{v},$ and the tail of $\vec{v}$ is a parallelogram. 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 $\ell$ maps every point $A$ in the plane into its image $A^{\prime }$ such that one of the following statements is satisfied.

• If $A$ is on the line $\ell ,$ then $A$ and $A^{\prime }$ are the same point.
• If $A$ is not on the line $\ell ,$ then $\ell$ is the perpendicular bisector of $\overline{A{A}^{\prime }}.$

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 ${\alpha }^{\circ }.$ The point $P$ is called the center of rotation. Rotations map every point $A$ in the plane to its image $A^{\prime }$ such that one of the following statements is satisfied.

• If $A$ is the center of rotation, then $A$ and $A^{\prime }$ are the same point.
• If $A$ is not the center of rotation, then $A$ and $A^{\prime }$ are the same distance from $P,$ and $\angle AP{A}^{\prime }$ has a measure of ${\alpha }^{\circ }.$

The angle formed by a preimage, the center of rotation, and the image is called the angle of rotation and its measure is ${\alpha }^{\circ }.$ Since rotations preserve side lengths and angle measures, they are rigid motions. 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.