Symmetry and Geometric Transformations

Vocabulary and rules based on lecture notes covering symmetry, rotational properties, and various types of geometric transformations including translations, reflections, rotations, and dilations.

Symmetry

An important concept in geometry meaning that an object looks the same when we fold, flip, slide, or turn it.

Asymmetric

A term used to describe an object if its two parts do not match exactly when evaluated for symmetry.

Reflection symmetry

Also called mirror symmetry, it happens when one half of an object is a mirror image of the other half.

Line of symmetry

Straight imaginary lines that divide a shape into two equal parts where one part is an exact reflection or mirror image of the other.

Rotational Symmetry

Happens when a shape or object still looks the same after it is rotated around a central point by a certain angle before completing a full $360^{\circ}$ rotation.

Center of Rotation

The fixed point around which a shape is turned; this point stays in the same position.

Angle of Rotation

The smallest angle at which a shape can be rotated to coincide with its original shape.

Geometric Transformations

Operations that change the position, size, or orientation of shapes while preserving their basic properties.

Transformation

An operation that moves, flips, or rotates a figure (the preimage) to create a new figure (the image).

Pre-image

The figure before transformation, often denoted as $P$.

Image

The figure after transformation, often denoted as $P'$ (prime).

Rigid Transformations

Transformations that preserve size and shape, also known as isometries.

Non-Rigid Transformations

Transformations that preserve shape but not size.

Translation

A rigid transformation often called a "slide" in which a shape is moved vertically, horizontally, or both, while every point moves the same distance in the same direction.

Coordinate Rule for Translation

$(x, y) \rightarrow (x+a, y+b)$ where $a$ is the horizontal shift and $b$ is the vertical shift.

Reflection

A rigid transformation often called a "flip" in which a shape is mirrored across a line called the line of reflection.

Rule for Reflection Over the x-axis

$$(x, y) \rightarrow (x, -y)$$

Rule for Reflection Over the y-axis

$$(x, y) \rightarrow (-x, y)$$

Rule for Reflection Over line $y=x$

$$(x, y) \rightarrow (y, x)$$

Rule for Reflection Over line $y=-x$

$$(x, y) \rightarrow (-y, -x)$$

Rotation

A rigid transformation often called a "turn" in which a shape is turned a certain number of degrees around a given point.

Rule for $90^{\circ}$ Counter clockwise Rotation (Origin)

$$(x, y) \rightarrow (-y, x)$$

Rule for $180^{\circ}$ Rotation (Origin)

$$(x, y) \rightarrow (-x, -y)$$

Rule for $275^{\circ}$ Counter clockwise Rotation (Origin)

$$(x, y) \rightarrow (y, -x)$$

Dilation

A non-rigid transformation often called a "resize" in which a shape is made bigger or smaller by a scale factor ($k$).

Enlargement

A dilation that creates a larger image.

Reduction

A dilation that creates a smaller image.

Scale Factor ($k$)

The factor used in dilation determined by the coordinate rule $(x, y) \rightarrow (kx, ky)$, indicating how much the shape is resized.