Congruence Transformations - Tenth Grade Geometry
Introduction to Congruence Transformations
Transformation: A change in position, size, or shape of a figure
Pre-image: The original figure before transformation
Image: The figure after transformation (denoted with prime notation: A')
Congruent: Same size and shape (side lengths and angle measures unchanged)
Congruence Transformation (Rigid Motion): A transformation that preserves size and shape
Pre-image: The original figure before transformation
Image: The figure after transformation (denoted with prime notation: A')
Congruent: Same size and shape (side lengths and angle measures unchanged)
Congruence Transformation (Rigid Motion): A transformation that preserves size and shape
1. Classify Congruence Transformations
Three Types of Congruence Transformations:
1. Translation (Slide) - Moving all points the same distance in same direction
2. Reflection (Flip) - Flipping figure over a line
3. Rotation (Turn) - Turning figure around a point
Key Property: All preserve congruence (size, shape, angles, side lengths)
Not Congruence Transformations: Dilation (resizing) - changes size
1. Translation (Slide) - Moving all points the same distance in same direction
2. Reflection (Flip) - Flipping figure over a line
3. Rotation (Turn) - Turning figure around a point
Key Property: All preserve congruence (size, shape, angles, side lengths)
Not Congruence Transformations: Dilation (resizing) - changes size
Properties of Congruence Transformations:
What is PRESERVED (stays the same):
• Length of sides
• Angle measures
• Perimeter
• Area
• Parallelism (parallel lines stay parallel)
• Collinearity (points on a line stay on a line)
What MAY CHANGE:
• Position (location)
• Orientation (which way figure faces)
Direct vs. Opposite Transformations:
• Direct (Preserves orientation): Translation, Rotation
• Opposite (Reverses orientation): Reflection
What is PRESERVED (stays the same):
• Length of sides
• Angle measures
• Perimeter
• Area
• Parallelism (parallel lines stay parallel)
• Collinearity (points on a line stay on a line)
What MAY CHANGE:
• Position (location)
• Orientation (which way figure faces)
Direct vs. Opposite Transformations:
• Direct (Preserves orientation): Translation, Rotation
• Opposite (Reverses orientation): Reflection
2. Translations (Slides)
Translation: A transformation that slides all points the same distance in the same direction
Vector: Shows direction and distance of translation
Notation: $(x, y) \to (x + h, y + k)$ where $h$ and $k$ are horizontal and vertical shifts
Result: Pre-image and image are congruent and have same orientation
Vector: Shows direction and distance of translation
Notation: $(x, y) \to (x + h, y + k)$ where $h$ and $k$ are horizontal and vertical shifts
Result: Pre-image and image are congruent and have same orientation
Translation Rules
Translation Formula:
$$T_{(h,k)}(x, y) = (x + h, y + k)$$
Where:
• $h$ = horizontal shift (positive = right, negative = left)
• $k$ = vertical shift (positive = up, negative = down)
Common Translations:
• Right: $(x, y) \to (x + h, y)$
• Left: $(x, y) \to (x - h, y)$
• Up: $(x, y) \to (x, y + k)$
• Down: $(x, y) \to (x, y - k)$
• Diagonal: $(x, y) \to (x + h, y + k)$
$$T_{(h,k)}(x, y) = (x + h, y + k)$$
Where:
• $h$ = horizontal shift (positive = right, negative = left)
• $k$ = vertical shift (positive = up, negative = down)
Common Translations:
• Right: $(x, y) \to (x + h, y)$
• Left: $(x, y) \to (x - h, y)$
• Up: $(x, y) \to (x, y + k)$
• Down: $(x, y) \to (x, y - k)$
• Diagonal: $(x, y) \to (x + h, y + k)$
Example 1: Graph the image
Translate point A(2, 3) using rule $(x, y) \to (x + 4, y - 2)$
$A(2, 3) \to A'(2 + 4, 3 - 2) = A'(6, 1)$
The point moves 4 units right and 2 units down
Translate point A(2, 3) using rule $(x, y) \to (x + 4, y - 2)$
$A(2, 3) \to A'(2 + 4, 3 - 2) = A'(6, 1)$
The point moves 4 units right and 2 units down
Example 2: Find coordinates
Triangle ABC with A(1, 2), B(3, 5), C(4, 1) is translated 3 units left and 2 units up. Find A'B'C'.
Translation rule: $(x, y) \to (x - 3, y + 2)$
$A(1, 2) \to A'(1-3, 2+2) = A'(-2, 4)$
$B(3, 5) \to B'(3-3, 5+2) = B'(0, 7)$
$C(4, 1) \to C'(4-3, 1+2) = C'(1, 3)$
Triangle ABC with A(1, 2), B(3, 5), C(4, 1) is translated 3 units left and 2 units up. Find A'B'C'.
Translation rule: $(x, y) \to (x - 3, y + 2)$
$A(1, 2) \to A'(1-3, 2+2) = A'(-2, 4)$
$B(3, 5) \to B'(3-3, 5+2) = B'(0, 7)$
$C(4, 1) \to C'(4-3, 1+2) = C'(1, 3)$
Example 3: Write the rule
Point P(5, 7) is translated to P'(2, 10). Write the translation rule.
$h = x' - x = 2 - 5 = -3$
$k = y' - y = 10 - 7 = 3$
Translation rule: $(x, y) \to (x - 3, y + 3)$
In words: 3 units left and 3 units up
Point P(5, 7) is translated to P'(2, 10). Write the translation rule.
$h = x' - x = 2 - 5 = -3$
$k = y' - y = 10 - 7 = 3$
Translation rule: $(x, y) \to (x - 3, y + 3)$
In words: 3 units left and 3 units up
3. Reflections (Flips)
Reflection: A transformation that flips a figure over a line of reflection
Line of Reflection: The "mirror line" - perpendicular bisector of segments connecting pre-image to image points
Key Property: Each point and its image are equidistant from line of reflection
Result: Pre-image and image are congruent but orientation is reversed
Line of Reflection: The "mirror line" - perpendicular bisector of segments connecting pre-image to image points
Key Property: Each point and its image are equidistant from line of reflection
Result: Pre-image and image are congruent but orientation is reversed
Reflection Rules
Common Reflection Formulas:
1. Reflection over x-axis:
$$r_{x-axis}(x, y) = (x, -y)$$
Keep x, change sign of y
2. Reflection over y-axis:
$$r_{y-axis}(x, y) = (-x, y)$$
Change sign of x, keep y
3. Reflection over line $y = x$:
$$r_{y=x}(x, y) = (y, x)$$
Swap x and y coordinates
4. Reflection over line $y = -x$:
$$r_{y=-x}(x, y) = (-y, -x)$$
Swap and change signs of both coordinates
5. Reflection over origin:
$$r_{origin}(x, y) = (-x, -y)$$
Change signs of both coordinates
1. Reflection over x-axis:
$$r_{x-axis}(x, y) = (x, -y)$$
Keep x, change sign of y
2. Reflection over y-axis:
$$r_{y-axis}(x, y) = (-x, y)$$
Change sign of x, keep y
3. Reflection over line $y = x$:
$$r_{y=x}(x, y) = (y, x)$$
Swap x and y coordinates
4. Reflection over line $y = -x$:
$$r_{y=-x}(x, y) = (-y, -x)$$
Swap and change signs of both coordinates
5. Reflection over origin:
$$r_{origin}(x, y) = (-x, -y)$$
Change signs of both coordinates
Example 1: Reflect over x-axis
Reflect point A(3, 5) over the x-axis
Using rule: $(x, y) \to (x, -y)$
$A(3, 5) \to A'(3, -5)$
The point is flipped to the opposite side of x-axis
Reflect point A(3, 5) over the x-axis
Using rule: $(x, y) \to (x, -y)$
$A(3, 5) \to A'(3, -5)$
The point is flipped to the opposite side of x-axis
Example 2: Reflect over y-axis
Triangle DEF with D(-2, 3), E(-4, 1), F(-1, 2) reflected over y-axis. Find D'E'F'.
Using rule: $(x, y) \to (-x, y)$
$D(-2, 3) \to D'(2, 3)$
$E(-4, 1) \to E'(4, 1)$
$F(-1, 2) \to F'(1, 2)$
Triangle DEF with D(-2, 3), E(-4, 1), F(-1, 2) reflected over y-axis. Find D'E'F'.
Using rule: $(x, y) \to (-x, y)$
$D(-2, 3) \to D'(2, 3)$
$E(-4, 1) \to E'(4, 1)$
$F(-1, 2) \to F'(1, 2)$
Example 3: Reflect over $y = x$
Reflect point B(4, 7) over line $y = x$
Using rule: $(x, y) \to (y, x)$
$B(4, 7) \to B'(7, 4)$
Coordinates are swapped
Reflect point B(4, 7) over line $y = x$
Using rule: $(x, y) \to (y, x)$
$B(4, 7) \to B'(7, 4)$
Coordinates are swapped
Example 4: Find coordinates
Point P(6, -3) is reflected over the origin. Find P'.
Using rule: $(x, y) \to (-x, -y)$
$P(6, -3) \to P'(-6, 3)$
Point P(6, -3) is reflected over the origin. Find P'.
Using rule: $(x, y) \to (-x, -y)$
$P(6, -3) \to P'(-6, 3)$
4. Rotations (Turns)
Rotation: A transformation that turns a figure around a fixed point
Center of Rotation: The fixed point around which figure rotates
Angle of Rotation: Number of degrees figure turns
Direction: Counterclockwise (positive) or Clockwise (negative)
Result: Pre-image and image are congruent, same orientation for even multiples of 180°
Center of Rotation: The fixed point around which figure rotates
Angle of Rotation: Number of degrees figure turns
Direction: Counterclockwise (positive) or Clockwise (negative)
Result: Pre-image and image are congruent, same orientation for even multiples of 180°
Rotation Rules (About Origin)
Common Rotation Formulas (Counterclockwise about origin):
1. Rotation 90° counterclockwise:
$$R_{90°}(x, y) = (-y, x)$$
2. Rotation 180° (either direction):
$$R_{180°}(x, y) = (-x, -y)$$
3. Rotation 270° counterclockwise (or 90° clockwise):
$$R_{270°}(x, y) = (y, -x)$$
4. Rotation 360°:
$$R_{360°}(x, y) = (x, y)$$
Returns to original position
Note: Clockwise rotations are negative angles
• 90° clockwise = -90° = 270° counterclockwise
• 180° clockwise = -180° = 180° (same either way)
1. Rotation 90° counterclockwise:
$$R_{90°}(x, y) = (-y, x)$$
2. Rotation 180° (either direction):
$$R_{180°}(x, y) = (-x, -y)$$
3. Rotation 270° counterclockwise (or 90° clockwise):
$$R_{270°}(x, y) = (y, -x)$$
4. Rotation 360°:
$$R_{360°}(x, y) = (x, y)$$
Returns to original position
Note: Clockwise rotations are negative angles
• 90° clockwise = -90° = 270° counterclockwise
• 180° clockwise = -180° = 180° (same either way)
Example 1: Rotate 90° counterclockwise
Rotate point A(3, 2) 90° counterclockwise about origin
Using rule: $(x, y) \to (-y, x)$
$A(3, 2) \to A'(-2, 3)$
Rotate point A(3, 2) 90° counterclockwise about origin
Using rule: $(x, y) \to (-y, x)$
$A(3, 2) \to A'(-2, 3)$
Example 2: Rotate 180°
Triangle PQR with P(1, 4), Q(3, 5), R(2, 1) rotated 180° about origin. Find P'Q'R'.
Using rule: $(x, y) \to (-x, -y)$
$P(1, 4) \to P'(-1, -4)$
$Q(3, 5) \to Q'(-3, -5)$
$R(2, 1) \to R'(-2, -1)$
Triangle PQR with P(1, 4), Q(3, 5), R(2, 1) rotated 180° about origin. Find P'Q'R'.
Using rule: $(x, y) \to (-x, -y)$
$P(1, 4) \to P'(-1, -4)$
$Q(3, 5) \to Q'(-3, -5)$
$R(2, 1) \to R'(-2, -1)$
Example 3: Rotate 270° counterclockwise
Rotate point B(4, -2) 270° counterclockwise about origin
Using rule: $(x, y) \to (y, -x)$
$B(4, -2) \to B'(-2, -4)$
Rotate point B(4, -2) 270° counterclockwise about origin
Using rule: $(x, y) \to (y, -x)$
$B(4, -2) \to B'(-2, -4)$
Example 4: Write the rule
Point M(5, 3) is rotated to M'(-3, 5). What is the rotation?
Compare: $(5, 3) \to (-3, 5)$
This matches pattern: $(x, y) \to (-y, x)$
Answer: 90° counterclockwise rotation about origin
Point M(5, 3) is rotated to M'(-3, 5). What is the rotation?
Compare: $(5, 3) \to (-3, 5)$
This matches pattern: $(x, y) \to (-y, x)$
Answer: 90° counterclockwise rotation about origin
5. Glide Reflections
Glide Reflection: A composition of a translation and a reflection
Components: Translation (glide) + Reflection
Key Property: Line of reflection is parallel to direction of translation
Order: Can reflect first then translate, OR translate first then reflect
Real-world Example: Footprints - alternating left and right steps
Result: Pre-image and image are congruent, orientation reversed
Components: Translation (glide) + Reflection
Key Property: Line of reflection is parallel to direction of translation
Order: Can reflect first then translate, OR translate first then reflect
Real-world Example: Footprints - alternating left and right steps
Result: Pre-image and image are congruent, orientation reversed
Glide Reflection Process:
Method 1: Translate then Reflect
Step 1: Apply translation $(x, y) \to (x + h, y + k)$
Step 2: Apply reflection over line parallel to translation direction
Method 2: Reflect then Translate
Step 1: Apply reflection over line
Step 2: Apply translation parallel to reflection line
Common Example:
Translate horizontally, then reflect over horizontal line:
$(x, y) \to (x + h, y) \to (x + h, -y)$
Or: $(x, y) \to (x, -y) \to (x + h, -y)$ (same result!)
Method 1: Translate then Reflect
Step 1: Apply translation $(x, y) \to (x + h, y + k)$
Step 2: Apply reflection over line parallel to translation direction
Method 2: Reflect then Translate
Step 1: Apply reflection over line
Step 2: Apply translation parallel to reflection line
Common Example:
Translate horizontally, then reflect over horizontal line:
$(x, y) \to (x + h, y) \to (x + h, -y)$
Or: $(x, y) \to (x, -y) \to (x + h, -y)$ (same result!)
Example 1: Glide reflection
Point A(2, 3) translated 4 units right, then reflected over x-axis. Find A'.
Step 1: Translate
$(2, 3) \to (2 + 4, 3) = (6, 3)$
Step 2: Reflect over x-axis
$(6, 3) \to (6, -3)$
Final image: A'(6, -3)
Point A(2, 3) translated 4 units right, then reflected over x-axis. Find A'.
Step 1: Translate
$(2, 3) \to (2 + 4, 3) = (6, 3)$
Step 2: Reflect over x-axis
$(6, 3) \to (6, -3)$
Final image: A'(6, -3)
Example 2: Glide reflection (reverse order)
Triangle with vertices B(1, 2). First reflect over x-axis, then translate 3 units right.
Step 1: Reflect over x-axis
$(1, 2) \to (1, -2)$
Step 2: Translate right 3
$(1, -2) \to (4, -2)$
Final image: B'(4, -2)
Triangle with vertices B(1, 2). First reflect over x-axis, then translate 3 units right.
Step 1: Reflect over x-axis
$(1, 2) \to (1, -2)$
Step 2: Translate right 3
$(1, -2) \to (4, -2)$
Final image: B'(4, -2)
6. Sequences of Congruence Transformations
Sequence (Composition): Two or more transformations performed in order
Notation: $(T_2 \circ T_1)(x, y)$ means do $T_1$ first, then $T_2$
Order Matters: Different orders can produce different results
Result: Final image is still congruent to pre-image
Notation: $(T_2 \circ T_1)(x, y)$ means do $T_1$ first, then $T_2$
Order Matters: Different orders can produce different results
Result: Final image is still congruent to pre-image
Common Sequences:
1. Translation followed by Translation:
• Results in another translation
• Add the horizontal and vertical shifts
2. Reflection followed by Reflection:
• Over parallel lines → Translation
• Over intersecting lines → Rotation
• Over same line → Returns to original
3. Rotation followed by Rotation (same center):
• Results in another rotation
• Add the angles
4. Translation + Reflection:
• Glide reflection (if parallel)
• Order may or may not matter
1. Translation followed by Translation:
• Results in another translation
• Add the horizontal and vertical shifts
2. Reflection followed by Reflection:
• Over parallel lines → Translation
• Over intersecting lines → Rotation
• Over same line → Returns to original
3. Rotation followed by Rotation (same center):
• Results in another rotation
• Add the angles
4. Translation + Reflection:
• Glide reflection (if parallel)
• Order may or may not matter
Example 1: Sequence of transformations
Point A(2, 1): Reflect over y-axis, then translate up 3 units. Find A'.
Step 1: Reflect over y-axis
$(2, 1) \to (-2, 1)$
Step 2: Translate up 3
$(-2, 1) \to (-2, 4)$
Final image: A'(-2, 4)
Point A(2, 1): Reflect over y-axis, then translate up 3 units. Find A'.
Step 1: Reflect over y-axis
$(2, 1) \to (-2, 1)$
Step 2: Translate up 3
$(-2, 1) \to (-2, 4)$
Final image: A'(-2, 4)
Example 2: Choose the sequence
Triangle ABC at (1,1), (3,1), (2,3) becomes A'B'C' at (-1,-1), (-3,-1), (-2,-3). What transformation?
Notice all coordinates changed signs: $(x, y) \to (-x, -y)$
This is 180° rotation about origin
(Could also be two reflections: over x-axis then y-axis)
Triangle ABC at (1,1), (3,1), (2,3) becomes A'B'C' at (-1,-1), (-3,-1), (-2,-3). What transformation?
Notice all coordinates changed signs: $(x, y) \to (-x, -y)$
This is 180° rotation about origin
(Could also be two reflections: over x-axis then y-axis)
Example 3: Find the rules
Point P(4, 2) transformed to P''(-2, -4) through two transformations. Describe.
Possible sequence 1:
Reflect over y-axis: $(4, 2) \to (-4, 2)$
Translate left 2, down 6: $(-4, 2) \to (-2, -4)$
Possible sequence 2:
Rotate 180°: $(4, 2) \to (-4, -2)$
Translate right 2, down 2: $(-4, -2) \to (-2, -4)$
Point P(4, 2) transformed to P''(-2, -4) through two transformations. Describe.
Possible sequence 1:
Reflect over y-axis: $(4, 2) \to (-4, 2)$
Translate left 2, down 6: $(-4, 2) \to (-2, -4)$
Possible sequence 2:
Rotate 180°: $(4, 2) \to (-4, -2)$
Translate right 2, down 2: $(-4, -2) \to (-2, -4)$
7. Transformations That Carry a Polygon Onto Itself
Symmetry: A transformation that maps a figure onto itself
Line of Symmetry: A reflection line that maps figure onto itself
Rotational Symmetry: A rotation (other than 360°) that maps figure onto itself
Order of Rotational Symmetry: Number of positions where figure looks the same during 360° rotation
Line of Symmetry: A reflection line that maps figure onto itself
Rotational Symmetry: A rotation (other than 360°) that maps figure onto itself
Order of Rotational Symmetry: Number of positions where figure looks the same during 360° rotation
Symmetries of Common Shapes:
Equilateral Triangle:
• 3 lines of symmetry
• Rotational symmetry: 120°, 240°
• Order: 3
Square:
• 4 lines of symmetry (2 diagonals, 2 through midpoints)
• Rotational symmetry: 90°, 180°, 270°
• Order: 4
Rectangle (not square):
• 2 lines of symmetry (through midpoints)
• Rotational symmetry: 180°
• Order: 2
Regular Pentagon:
• 5 lines of symmetry
• Rotational symmetry: 72°, 144°, 216°, 288°
• Order: 5
Regular n-gon:
• n lines of symmetry
• Rotational symmetry: $\frac{360°}{n}$, $\frac{2 \times 360°}{n}$, etc.
• Order: n
Equilateral Triangle:
• 3 lines of symmetry
• Rotational symmetry: 120°, 240°
• Order: 3
Square:
• 4 lines of symmetry (2 diagonals, 2 through midpoints)
• Rotational symmetry: 90°, 180°, 270°
• Order: 4
Rectangle (not square):
• 2 lines of symmetry (through midpoints)
• Rotational symmetry: 180°
• Order: 2
Regular Pentagon:
• 5 lines of symmetry
• Rotational symmetry: 72°, 144°, 216°, 288°
• Order: 5
Regular n-gon:
• n lines of symmetry
• Rotational symmetry: $\frac{360°}{n}$, $\frac{2 \times 360°}{n}$, etc.
• Order: n
Example 1: Find symmetries
List all transformations that carry a square ABCD onto itself.
Reflections (4):
• Over vertical line through center
• Over horizontal line through center
• Over diagonal AC
• Over diagonal BD
Rotations (3) about center:
• 90° counterclockwise
• 180°
• 270° counterclockwise
Identity (1): 360° rotation (or 0°)
Total: 8 symmetries
List all transformations that carry a square ABCD onto itself.
Reflections (4):
• Over vertical line through center
• Over horizontal line through center
• Over diagonal AC
• Over diagonal BD
Rotations (3) about center:
• 90° counterclockwise
• 180°
• 270° counterclockwise
Identity (1): 360° rotation (or 0°)
Total: 8 symmetries
Example 2: Rotational symmetry
A regular hexagon has what rotational symmetries?
Angle = $\frac{360°}{6} = 60°$
Rotational symmetries:
• 60° (order 1)
• 120° (order 2)
• 180° (order 3)
• 240° (order 4)
• 300° (order 5)
• 360° (order 6 - identity)
Order of rotational symmetry: 6
A regular hexagon has what rotational symmetries?
Angle = $\frac{360°}{6} = 60°$
Rotational symmetries:
• 60° (order 1)
• 120° (order 2)
• 180° (order 3)
• 240° (order 4)
• 300° (order 5)
• 360° (order 6 - identity)
Order of rotational symmetry: 6
Transformation Rules Quick Reference
| Transformation | Rule | Description |
|---|---|---|
| Translation | $(x, y) \to (x + h, y + k)$ | Slide h units horizontally, k units vertically |
| Reflection over x-axis | $(x, y) \to (x, -y)$ | Flip over x-axis |
| Reflection over y-axis | $(x, y) \to (-x, y)$ | Flip over y-axis |
| Reflection over $y = x$ | $(x, y) \to (y, x)$ | Swap coordinates |
| Reflection over $y = -x$ | $(x, y) \to (-y, -x)$ | Swap and negate coordinates |
| Rotation 90° CCW | $(x, y) \to (-y, x)$ | Rotate counterclockwise about origin |
| Rotation 180° | $(x, y) \to (-x, -y)$ | Rotate 180° about origin |
| Rotation 270° CCW | $(x, y) \to (y, -x)$ | Rotate 270° CCW (or 90° CW) |
Properties Comparison
| Property | Translation | Reflection | Rotation |
|---|---|---|---|
| Preserves size | Yes | Yes | Yes |
| Preserves shape | Yes | Yes | Yes |
| Preserves orientation | Yes (direct) | No (opposite) | Yes (direct) |
| Preserves angle measures | Yes | Yes | Yes |
| Preserves distance | Yes | Yes | Yes |
| Changes position | Yes | Yes | Yes |
Symmetries of Regular Polygons
| Polygon | Sides | Lines of Symmetry | Rotational Symmetry Angles | Order |
|---|---|---|---|---|
| Equilateral Triangle | 3 | 3 | 120°, 240° | 3 |
| Square | 4 | 4 | 90°, 180°, 270° | 4 |
| Regular Pentagon | 5 | 5 | 72°, 144°, 216°, 288° | 5 |
| Regular Hexagon | 6 | 6 | 60°, 120°, 180°, 240°, 300° | 6 |
| Regular Octagon | 8 | 8 | 45°, 90°, 135°, 180°, 225°, 270°, 315° | 8 |
| Regular n-gon | n | n | $\frac{360°}{n}$, $\frac{2 \times 360°}{n}$, ... | n |
Key Formulas Summary
| Concept | Formula | Notes |
|---|---|---|
| Translation Vector | $\langle h, k \rangle$ or $(h, k)$ | h = horizontal, k = vertical shift |
| Distance Preserved | $d(A, B) = d(A', B')$ | True for all congruence transformations |
| Angle Preserved | $m\angle ABC = m\angle A'B'C'$ | True for all congruence transformations |
| Regular n-gon Rotation | $\frac{360°}{n}$ | Minimum angle of rotational symmetry |
| Glide Reflection | Translation + Reflection | Line of reflection ∥ direction of translation |
Success Tips for Congruence Transformations:
✓ Congruence transformations: Translation, Reflection, Rotation (preserve size and shape)
✓ Translation: $(x, y) \to (x + h, y + k)$ - slides figure
✓ Reflection over x-axis: $(x, y) \to (x, -y)$; over y-axis: $(x, y) \to (-x, y)$
✓ Rotation 90° CCW: $(x, y) \to (-y, x)$; 180°: $(x, y) \to (-x, -y)$
✓ Glide reflection = translation + reflection (or vice versa)
✓ Order matters in sequences (except for some cases)
✓ Reflections reverse orientation; translations and rotations preserve it
✓ Regular n-gon has n lines of symmetry and rotational symmetry of $\frac{360°}{n}$
✓ Prime notation (A') indicates image after transformation
✓ All congruence transformations preserve distance, angles, and parallelism!
✓ Congruence transformations: Translation, Reflection, Rotation (preserve size and shape)
✓ Translation: $(x, y) \to (x + h, y + k)$ - slides figure
✓ Reflection over x-axis: $(x, y) \to (x, -y)$; over y-axis: $(x, y) \to (-x, y)$
✓ Rotation 90° CCW: $(x, y) \to (-y, x)$; 180°: $(x, y) \to (-x, -y)$
✓ Glide reflection = translation + reflection (or vice versa)
✓ Order matters in sequences (except for some cases)
✓ Reflections reverse orientation; translations and rotations preserve it
✓ Regular n-gon has n lines of symmetry and rotational symmetry of $\frac{360°}{n}$
✓ Prime notation (A') indicates image after transformation
✓ All congruence transformations preserve distance, angles, and parallelism!