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Reference

Rigid Motions Properties and Examples

Concept

Rigid Motion

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.
Two logos of Mathleaks with the letters ML and points A, B and their images A' and B'
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.

Rule

Properties of Rigid Motions

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.

Rigid motions applied to a polygon

Proof

  • 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.

Concept

Translation of Geometric Objects

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 maps every point in the plane onto its image such that the following statements hold true.

Notice that these three properties imply that the quadrilateral formed by the tip of and the tail of is a parallelogram.
Vector v, point A and its image A under a translation along v
Since translations preserve side lengths and angle measures, they are rigid motions. Additionally, translations map lines onto parallel lines.

Concept

Reflection of Geometric Objects

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.
Triangle being reflected across a movable line
More precisely, a reflection across a line maps every point in the plane into its image such that one of the following statements is satisfied.
  • If is on the line then and are the same point.
  • If is not on the line then is the perpendicular bisector of
Segment AA' intersects line ell perpendicularly, and line ell bisects segment AA'. Points B and B' coincide.

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.

Concept

Rotation

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

  • If is the center of rotation, then and are the same point.
  • If is not the center of rotation, then and are the same distance from and has a measure of
The angle formed by a preimage, the center of rotation, and the image is called the angle of rotation and its measure is Since rotations preserve side lengths and angle measures, they are rigid motions.
Rotation of point A around center P
Usually, rotations are performed counterclockwise unless otherwise stated.

Concept

Glide Reflection

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.
Performing a Glide Reflection on a Triangle
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.
Performing a Glide Reflection on a Triangle