Rotations, Reflections
& Translations
Develop precise definitions of the three rigid motions using angles, circles, perpendicular lines, parallel lines, and line segments.
Geometry
What Are Rigid Motions?
A rigid motion (also called an isometry) is a transformation of the plane that preserves distances and angle measures. Every point moves, but the shape and size of every figure remain exactly the same.
Translation
Slides every point the same distance in the same direction
Reflection
Flips every point across a fixed line of reflection
Rotation
Turns every point around a fixed center by a given angle
Why Do Precise Definitions Matter?
Saying "a translation slides a shape" is informal. To prove theorems in geometry, we need definitions built from measurable objects: distances, angles, perpendicular lines, and parallel lines.
1
Parallel lines define the direction of a translation.
2
Perpendicular bisectors define the mirror line of a reflection.
3
Circles and angles define the arc of a rotation.
Defining a Translation
Translation: Key Properties
1
Parallel segments: Every segment \( \overline{PP'} \) is parallel to the translation vector.
2
Equal length: \( PP' \) is the same for every point \( P \).
3
No fixed points: Unless the translation is the identity (zero vector), no point maps to itself.
4
Preserves orientation: The image has the same clockwise/counterclockwise order of vertices.
Defining a Reflection
Reflection: Key Properties
1
Perpendicularity: \( \overline{PP'} \perp m \) for every point \( P \) not on \( m \).
2
Equal distance: The distance from \( P \) to \( m \) equals the distance from \( P' \) to \( m \).
3
Fixed points: Every point on line \( m \) maps to itself.
4
Reverses orientation: Clockwise vertex order becomes counterclockwise (and vice versa).
Defining a Rotation
Rotation: Key Properties
1
Circles: Every point \( P \) traces an arc of a circle centered at \( O \) with radius \( OP \).
2
Constant angle: The angle \( \angle POP' \) is the same for every point.
3
One fixed point: The center \( O \) maps to itself. No other point is fixed (unless \( \theta = 0^\circ \) or \( 360^\circ \)).
4
Preserves orientation: Like translations, vertex order stays the same.
Comparing the Three Rigid Motions
| Property | Translation | Reflection | Rotation |
|---|---|---|---|
| Defined by | Direction vector | Line of reflection | Center + angle |
| Fixed points | None | Points on line | Center only |
| Orientation | Preserved | Reversed | Preserved |
| Key geometry | Parallel lines | Perpendicular bisector | Circles, angles |
| Preserves distance | Yes | Yes | Yes |
How to Identify a Transformation
1
Check orientation: Label vertices in order. If the order flips (CW to CCW), it is a reflection.
2
Look for fixed points: If one point stays put, it may be the center of rotation.
3
Check parallel segments: If every \( \overline{PP'} \) is parallel and equal, it is a translation.
4
Measure distances to a center: If \( OP = OP' \) for all points and orientation is preserved, it is a rotation.
5
Verify perpendicular bisector: If a single line is the perpendicular bisector of every \( \overline{PP'} \), it is a reflection.
Example 1: Identifying a Translation
Example 2: Identifying a Reflection
Example 3: Identifying a Rotation
Common Mistakes
Confusing Rotation Direction
A positive angle is counterclockwise. Writing \( 90^\circ \) CW when you mean \( -90^\circ \) (or \( 270^\circ \) CCW) gives the wrong image.
Forgetting the Line of Reflection
A reflection requires a specific line. Reflecting over the x-axis vs. the y-axis produces completely different images.
Claiming Reflections Preserve Orientation
Only translations and rotations preserve orientation. Reflections reverse it: if vertices go A-B-C clockwise, the image goes A'-B'-C' counterclockwise.
Summary of Formal Definitions
Translation
Every \( \overline{PP'} \) is parallel and equal in length: defined by a direction vector along parallel lines.
Reflection
Line \( m \) is the perpendicular bisector of every \( \overline{PP'} \): defined by a line, using perpendicularity.
Rotation
\( OP = OP' \) and \( m\angle POP' = \theta \): defined by a center, an angle, and arcs of circles.
Question 1 of 5
A transformation maps every point \( P \) to \( P' \) so that the line \( \ell \) is the perpendicular bisector of \( \overline{PP'} \). Which transformation is this?
The defining property of a reflection is that the line of reflection is the perpendicular bisector of every segment joining a point to its image.
Question 2 of 5
In a translation, the segments \( \overline{AA'} \) and \( \overline{BB'} \) (connecting each point to its image) are always:
By definition, a translation moves every point the same distance in the same direction, so \( \overline{AA'} \parallel \overline{BB'} \) and \( AA' = BB' \).
Question 3 of 5
A rotation of \( 90^\circ \) counterclockwise about the origin maps \( (4, 0) \) to which point?
A \( 90^\circ \) CCW rotation uses the rule \( (x,y) \to (-y, x) \). Applying to \( (4,0) \): \( (-0, 4) = (0, 4) \). The point stays on the circle of radius 4 but turns \( 90^\circ \) upward.
Question 4 of 5
Which rigid motion reverses the orientation of a figure?
Translations and rotations preserve orientation (the vertex order stays the same). Only a reflection reverses it, swapping clockwise and counterclockwise order.
Question 5 of 5
A transformation has exactly one fixed point and preserves distances. It must be a:
A non-identity translation has no fixed points. A reflection fixes every point on the mirror line (infinitely many). A rotation (other than \( 0^\circ \) or \( 360^\circ \)) fixes only the center. A dilation does not preserve distances.