Geometry Unit 2A

Rigid Transformations (Preserve Congruence): Translations (slides), reflections (flips), and rotations (turns) keep shapes identical in size and angle measure.
Non-Rigid Transformations (Dilations): Changing the scale factor (e.g., scale factor of $2$ or $0.5$ ) resizes the shape, making it similar rather than congruent.

Across $x$ -axis ( $y = 0$ ): $(x, y) \rightarrow (x, -y)$
Across $y$ -axis ( $x = 0$ ): $(x, y) \rightarrow (-x, y)$
Across line $y = x$ : $(x, y) \rightarrow (y, x)$
Across a vertical line ( $x = c$ ):
- Only the $x$ -coordinate changes ( $y$ stays the same).
- Count horizontal distance from your $x$ -coordinate to the line $x = c$ , then count that same distance past the line on the other side.
- Shortcut formula: $(x, y) \rightarrow (2c - x, y)$
- Quick Example: Reflect $(-4, 3)$ across $x = 1$ . Distance from $-4$ to $1$ is $5$ units right. Move $5$ more units right past $1 \rightarrow (6, 3)$ .
Across a horizontal line ( $y = c$ ):
- Only the $y$ -coordinate changes ( $x$ stays the same).
- Count vertical distance from your $y$ -coordinate to the line $y = c$ , then count that same distance past the line on the other side.
- Shortcut formula: $(x, y) \rightarrow (x, 2c - y)$
- Quick Example: Reflect $(-3, 5)$ across $y = 1$ . Distance from $5$ to $1$ is $4$ units down. Move $4$ more units down past $1 \rightarrow (-3, -3)$ .

$90^\circ$ Counterclockwise / $270^\circ$ Clockwise: $(x, y) \rightarrow (-y, x)$
$180^\circ$ (Clockwise or Counterclockwise): $(x, y) \rightarrow (-x, -y)$
$90^\circ$ Clockwise / $270^\circ$ Counterclockwise: $(x, y) \rightarrow (y, -x)$

Shift right/left by $a$ and up/down by $b$ : $(x, y) \rightarrow (x \pm a, y \pm b)$

Lines of Symmetry: A regular $n$ -sided polygon always has exactly $n$ lines of symmetry (e.g., a $10$ -sided decagon has $10$ lines of symmetry).
Rotational Symmetry Angle: Calculate $\frac{360^\circ}{n}$ to find the smallest angle that maps a regular polygon onto itself (e.g., $\frac{360^\circ}{6} = 60^\circ$ for a regular hexagon; $\frac{360^\circ}{10} = 36^\circ$ for a regular decagon).

Midpoint Formula: $\left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)$ - Use this to check if diagonals bisect each other by sharing the exact same center midpoint.
Distance Formula: $d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$ - Use this to prove line segments are equal in length (congruent).
Slope Formula: $m = \frac{y2 - y1}{x2 - x1}$ - Parallel lines have equal slopes ( $m1 = m2$ ); perpendicular lines have negative reciprocal slopes (e.g., $2$ and $-\frac{1}{2}$ ).