Reflections

Reflections, also known as flips, create mirror images of figures across a line of reflection.

Definition: A reflection is a transformation that flips a figure over a line to create a mirror image.

In the coordinate plane, reflections can be performed over the x-axis or y-axis:

Example:

• Reflection over x-axis: (x, y) → $x, -y$
• Reflection over y-axis: (x, y) → $-x, y$

Highlight: In a reflection, the original figure and its image are identical.

Rotations

Rotations involve turning figures around a point called the center of rotation by a specific angle.

Definition: A rotation is a transformation that turns a figure around a fixed point by a certain angle.

In the coordinate plane, rotations about the origin follow specific rules:

Example:

• 90° counterclockwise: (x, y) → $-y, x$
• 180° rotation: (x, y) → $-x, -y$
• 270° counterclockwise: (x, y) → $y, -x$

Highlight: In a rotation, the original figure and its image are identical.

Congruent Figures

Congruent figures have the same size and shape, and can be transformed into each other through rigid motions.

Definition: Rigid motions are transformations that preserve length and angle measure, including translations, reflections, and rotations.

Vocabulary: Congruent figures have congruent angles and congruent sides.

Example: In congruent triangles, corresponding sides and angles are equal.

Dilations

Dilations change the size of figures while maintaining their shape, with respect to a center of dilation.

Definition: A dilation is a transformation that enlarges or reduces a figure with respect to a fixed point.

In the coordinate plane, dilations are performed by multiplying coordinates by a scale factor:

Example: (x, y) → (kx, ky)

Where k > 1 results in enlargement, and 0 < k < 1 results in reduction.

Highlight: In a dilation, the ratio of side lengths of the image to the original figure is equal to the scale factor.

Similar Figures

Similar figures have the same shape but not necessarily the same size, and can be transformed into each other through similarity transformations.

Definition: A similarity transformation is a sequence of dilations and rigid motions.

Highlight: Corresponding angles of similar figures are congruent, and corresponding side lengths are proportional.

Perimeters and Areas of Similar Figures

The relationship between perimeters and areas of similar figures is based on their scale factor.

Highlight: The ratio of perimeters of similar figures is equal to the ratio of their corresponding side lengths.

Example: For similar triangles ABC and DEF: Perimeter of ABC / Perimeter of DEF = AB/DE = BC/EF = AC/DF

Highlight: The ratio of areas of similar figures is equal to the square of the ratio of their corresponding side lengths.

Example: For similar triangles ABC and DEF: Area of ABC / Area of DEF = (AB/DE)² = (BC/EF)² = (AC/DF)²

Translations

Translations are transformations where figures slide without turning. Every point moves the same distance and direction.

Definition: A translation is a transformation that slides a figure without rotating it.

In the coordinate plane, translations are performed by adding values to x and y coordinates:

Example: (x, y) → $x + a, y + b$

Where a represents horizontal movement and b represents vertical movement. Positive values indicate up/right translations, while negative values indicate down/left translations.

Highlight: In a translation, the original figure and its image are identical.