Rigid Transformations on the Coordinate Plane

A rigid transformation is a movement of a figure on a coordinate plane that does not change its size or shape. Because the original figure and the transformed figure (the image) are identical in size and shape, they are congruent.

There are three main types of rigid transformations: translations, reflections, and rotations.

Translations (Sliding)

A translation slides a figure horizontally, vertically, or both, without turning or flipping it. We describe translations using a rule: $(x, y) \rightarrow (x + a, y + b)$ where $a$ is the horizontal shift and $b$ is the vertical shift.

Example: Translate a triangle with vertices $(1,2)$, $(3,4)$, and $(5,1)$ by the vector $(-2, 3)$.

• $(1, 2) \rightarrow (1 - 2, 2 + 3) = (-1, 5)$
• $(3, 4) \rightarrow (3 - 2, 4 + 3) = (1, 7)$
• $(5, 1) \rightarrow (5 - 2, 1 + 3) = (3, 4)$

Reflections (Flipping)

A reflection flips a figure over a specific line, creating a mirror image. Common reflection rules include:

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

Example: Reflect a quadrilateral with a vertex at $(4, -2)$ over the line $y = x$. Using the rule $(x, y) \rightarrow (y, x)$, the new vertex is $(-2, 4)$.

Rotations (Turning)

A rotation turns a figure around a fixed point, usually the origin $(0,0)$. Standard rules for counterclockwise (CCW) rotations about the origin are:

• $90^\circ$ CCW: $(x, y) \rightarrow (-y, x)$
• $180^\circ$: $(x, y) \rightarrow (-x, -y)$
• $270^\circ$ CCW (or $90^\circ$ CW): $(x, y) \rightarrow (y, -x)$

Example: Rotate a figure with a vertex at $(3, 5)$ by $90^\circ$ counterclockwise about the origin. Using the rule $(x, y) \rightarrow (-y, x)$, the point $(3, 5)$ becomes $(-5, 3)$.