Understanding Rigid Motions

In geometry, a rigid motion (or isometry) is a transformation that preserves distances and angle measures. Because the size and shape of the figure do not change, the original figure (pre-image) and the new figure (image) are always congruent.

There are three basic types of rigid motions on the coordinate plane: translations, reflections, and rotations.

1. Translations (Slides)

A translation moves every point of a figure the same distance in the same direction. It is described by adding or subtracting values from the $x$ and $y$ coordinates.

• Coordinate Rule: $(x, y) \to (x + a, y + b)$
• Here, $a$ represents the horizontal shift and $b$ represents the vertical shift.

2. Reflections (Flips)

A reflection flips a figure over a specific line, called the line of reflection. This creates a mirror image.

Common coordinate rules for reflections include:

• Over the $x$-axis: $(x, y) \to (x, -y)$
• Over the $y$-axis: $(x, y) \to (-x, y)$
• Over the line $y = x$: $(x, y) \to (y, x)$
• Over the line $y = -x$: $(x, y) \to (-y, -x)$

3. Rotations (Turns)

A rotation turns a figure around a fixed point, usually the origin $(0, 0)$. Unless stated otherwise, rotations are assumed to be counterclockwise.

Common coordinate rules for rotations about the origin:

• $90^\circ$ rotation: $(x, y) \to (-y, x)$
• $180^\circ$ rotation: $(x, y) \to (-x, -y)$
• $270^\circ$ rotation: $(x, y) \to (y, -x)$

Compositions of Transformations

A composition occurs when two or more transformations are performed in sequence. The output of the first transformation becomes the input for the second.

Example: Find the image of the point $(3, 5)$ after a reflection over $y = x$ followed by a translation of $(2, -1)$.

1. First transformation (Reflection over $y = x$): Apply the rule $(x, y) \to (y, x)$. The point $(3, 5)$ becomes $(5, 3)$.
2. Second transformation (Translation): Apply the rule $(x, y) \to (x + 2, y - 1)$ to our new point. The point $(5, 3)$ becomes $(5 + 2, 3 - 1) = (7, 2)$.

The final image is $(7, 2)$.

Describing Transformations

If you are given the coordinates of a pre-image $\triangle ABC$ and its image $\triangle A'B'C'$, you can determine the rigid motion by looking for coordinate patterns. For example, if $A(1, 4)$ moves to $A'(-4, 1)$, you can see the pattern $(x, y) \to (-y, x)$, which tells you the transformation was a $90^\circ$ counterclockwise rotation.