1. Identify Reflections, Rotations, and Translations

Transformations: Changes to a figure's position, size, or orientation in the coordinate plane.

Three Rigid Transformations (Preserve Size and Shape):

1. Translation (Slide):

• Moves every point the same distance in the same direction
• Like sliding a shape without turning it
• No change in orientation
• Example: Moving a triangle 3 units right and 2 units up

2. Reflection (Flip):

• Flips a figure over a line (line of reflection)
• Creates a mirror image
• Changes orientation
• Example: Flipping over the x-axis or y-axis

3. Rotation (Turn):

• Turns a figure around a fixed point (center of rotation)
• Measured in degrees
• Can be clockwise or counterclockwise
• Example: Rotating 90° counterclockwise around the origin

Key Properties:

• Rigid transformations preserve distance and angle measures
• Pre-image: Original figure before transformation
• Image: Resulting figure after transformation (denoted with prime notation: A')
• Isometry: A transformation that preserves distance

2. Translations: Rules and Coordinates

Translation Rule:

\( (x, y) \rightarrow (x + a, y + b) \)

where \( a \) = horizontal shift (positive = right, negative = left)

where \( b \) = vertical shift (positive = up, negative = down)

Notation:

Vector notation: \( \langle a, b \rangle \) or \( T_{a,b} \)

Examples:

Example 1: Translate point \( A(3, 4) \) by 5 units right and 2 units down.

Rule: \( (x, y) \rightarrow (x + 5, y - 2) \)

\( A(3, 4) \rightarrow A'(3 + 5, 4 - 2) = A'(8, 2) \)

Example 2: Write the rule for translating 4 units left and 3 units up.

Rule: \( (x, y) \rightarrow (x - 4, y + 3) \)

Example 3: Point \( B(2, -3) \) translates to \( B'(5, 1) \). Find the translation rule.

Horizontal shift: \( 5 - 2 = 3 \) (right)

Vertical shift: \( 1 - (-3) = 4 \) (up)

Rule: \( (x, y) \rightarrow (x + 3, y + 4) \)

3. Reflections: Rules and Coordinates

Reflection Rules:

Examples:

Example 1: Reflect point \( P(4, 3) \) over the x-axis.

Rule: \( (x, y) \rightarrow (x, -y) \)

\( P(4, 3) \rightarrow P'(4, -3) \)

Example 2: Reflect point \( Q(-2, 5) \) over the y-axis.

Rule: \( (x, y) \rightarrow (-x, y) \)

\( Q(-2, 5) \rightarrow Q'(2, 5) \)

Example 3: Reflect point \( R(3, -4) \) over the line \( y = x \).

Rule: \( (x, y) \rightarrow (y, x) \)

\( R(3, -4) \rightarrow R'(-4, 3) \)

Example 4: Reflect point \( S(6, 2) \) over the origin.

Rule: \( (x, y) \rightarrow (-x, -y) \)

\( S(6, 2) \rightarrow S'(-6, -2) \)

4. Rotations: Rules and Coordinates

Convention: Positive angles = counterclockwise, Negative angles = clockwise

Rotation Rules (About the Origin):

Examples:

Example 1: Rotate point \( A(3, 4) \) 90° counterclockwise about the origin.

Rule: \( (x, y) \rightarrow (-y, x) \)

\( A(3, 4) \rightarrow A'(-4, 3) \)

Example 2: Rotate point \( B(5, -2) \) 180° about the origin.

Rule: \( (x, y) \rightarrow (-x, -y) \)

\( B(5, -2) \rightarrow B'(-5, 2) \)

Example 3: Rotate point \( C(-3, 6) \) 270° counterclockwise about the origin.

Rule: \( (x, y) \rightarrow (y, -x) \)

\( C(-3, 6) \rightarrow C'(6, 3) \)

Example 4: Rotate point \( D(2, 7) \) 90° clockwise about the origin.

Rule: \( (x, y) \rightarrow (y, -x) \)

\( D(2, 7) \rightarrow D'(7, -2) \)

5. Describe a Sequence of Transformations

Definition: A composition of transformations is when two or more transformations are applied one after another.

Important Notes:

• Order matters! Different orders can produce different results
• Apply transformations in the order given
• The image from one transformation becomes the pre-image for the next
• Use prime notation: A → A' → A''

Examples:

Example 1: Point \( P(2, 3) \) is reflected over the x-axis, then translated 4 units right.

Step 1: Reflect over x-axis: \( P(2, 3) \rightarrow P'(2, -3) \)

Step 2: Translate 4 right: \( P'(2, -3) \rightarrow P''(6, -3) \)

Final image: \( P''(6, -3) \)

Example 2: Point \( Q(4, 1) \) is rotated 90° CCW, then reflected over the y-axis.

Step 1: Rotate 90° CCW: \( Q(4, 1) \rightarrow Q'(-1, 4) \)

Step 2: Reflect over y-axis: \( Q'(-1, 4) \rightarrow Q''(1, 4) \)

Final image: \( Q''(1, 4) \)

Example 3: Describe the transformations that map \( A(2, 3) \) to \( A''(2, -1) \).

Possible sequence: Translate 4 units down: \( (x, y) \rightarrow (x, y - 4) \)

Or: Reflect over x-axis, then translate 2 units down

Special Cases:

Two reflections over parallel lines = Translation

Two reflections over intersecting lines = Rotation (angle = 2 × angle between lines)

6. Identify Congruent Figures

Definition: Two figures are congruent if they have the same size and shape.

Congruence Symbol:

\( \triangle ABC \cong \triangle DEF \)

Read as: "Triangle ABC is congruent to triangle DEF"

Properties of Congruent Figures:

• Corresponding sides are equal in length
• Corresponding angles are equal in measure
• Can be mapped onto each other using rigid transformations
• Area and perimeter are equal

Congruence Transformations:

Rigid transformations that preserve congruence:

• Translation: Preserves size, shape, and orientation
• Reflection: Preserves size and shape; reverses orientation
• Rotation: Preserves size, shape, and changes orientation

How to Prove Congruence:

1. Show that one figure can be mapped to another using only rigid transformations
2. Verify that all corresponding sides are equal
3. Verify that all corresponding angles are equal

Example:

If \( \triangle ABC \cong \triangle DEF \), and \( AB = 5 \), \( BC = 7 \), \( \angle B = 60° \), find \( DE \), \( EF \), and \( \angle E \).

Corresponding parts: \( AB \leftrightarrow DE \), \( BC \leftrightarrow EF \), \( \angle B \leftrightarrow \angle E \)

\( DE = 5 \), \( EF = 7 \), \( \angle E = 60° \)

7. Congruence Statements and Corresponding Parts

Congruence Statement: A statement that identifies which parts of two figures correspond.

Important Rules:

• Order matters! Vertices must be listed in corresponding order
• First vertex corresponds to first vertex, second to second, etc.
• CPCTC: Corresponding Parts of Congruent Triangles are Congruent

Examples:

Example 1: If \( \triangle PQR \cong \triangle STU \), list all corresponding parts.

Corresponding sides:

• \( PQ \cong ST \)
• \( QR \cong TU \)
• \( RP \cong US \)

Corresponding angles:

• \( \angle P \cong \angle S \)
• \( \angle Q \cong \angle T \)
• \( \angle R \cong \angle U \)

Example 2: Given \( \triangle ABC \cong \triangle XYZ \). If \( AB = 8 \), \( \angle C = 45° \), and \( XY = 10 \), find the corresponding measures.

\( AB \) corresponds to \( XY \), so they should be equal

But \( AB = 8 \) and \( XY = 10 \) → NOT congruent! (contradiction)

\( \angle C \) corresponds to \( \angle Z \), so \( \angle Z = 45° \)

Example 3: Write a congruence statement for two congruent quadrilaterals where vertices A, B, C, D correspond to vertices W, X, Y, Z respectively.

Statement: \( ABCD \cong WXYZ \)

8. Side Lengths and Angle Measures of Congruent Figures

Key Principle: Corresponding parts of congruent figures are equal.

Steps to Find Missing Measures:

1. Identify the congruence statement
2. Determine which parts correspond
3. Set corresponding measures equal to each other
4. Solve for unknown values

Examples:

Example 1: \( \triangle ABC \cong \triangle DEF \). If \( AB = 12 \), \( BC = 15 \), \( CA = 9 \), find \( DE \), \( EF \), and \( FD \).

Corresponding sides:

\( AB \leftrightarrow DE \) → \( DE = 12 \)

\( BC \leftrightarrow EF \) → \( EF = 15 \)

\( CA \leftrightarrow FD \) → \( FD = 9 \)

Example 2: Quadrilateral ABCD ≅ Quadrilateral PQRS. If \( \angle A = 85° \), \( \angle B = 95° \), \( \angle C = 100° \), find \( \angle P \), \( \angle Q \), \( \angle R \), and \( \angle S \).

\( \angle P = 85° \) (corresponds to \( \angle A \))

\( \angle Q = 95° \) (corresponds to \( \angle B \))

\( \angle R = 100° \) (corresponds to \( \angle C \))

\( \angle S = 80° \) (sum of quadrilateral angles = 360°)

Example 3: \( \triangle JKL \cong \triangle MNO \). If \( JK = 2x + 3 \), \( MN = 5x - 6 \), find \( x \) and \( JK \).

Since \( JK \leftrightarrow MN \): \( 2x + 3 = 5x - 6 \)

\( 3 + 6 = 5x - 2x \)

\( 9 = 3x \) → \( x = 3 \)

\( JK = 2(3) + 3 = 9 \)

9. Determine if Two Figures are Congruent: Justify Your Answer

Methods to Justify Congruence:

Method 1: Transformation Approach

Show that one figure can be mapped to the other using a sequence of rigid transformations (translations, reflections, rotations).

Method 2: Measurement Approach

Verify that all corresponding sides and angles are equal.

Method 3: Triangle Congruence Theorems

• SSS: All three sides equal
• SAS: Two sides and included angle equal
• ASA: Two angles and included side equal
• AAS: Two angles and non-included side equal
• HL: Hypotenuse and leg of right triangles equal

Example Justifications:

Example 1: Triangle ABC and Triangle DEF have sides: AB = DE = 5, BC = EF = 7, CA = FD = 9. Are they congruent? Justify.

Yes, they are congruent.

Justification: All three corresponding sides are equal (SSS Congruence).

Example 2: Figure ABCD can be mapped to Figure WXYZ by a reflection over the y-axis followed by a translation 3 units up. Are they congruent?

Yes, they are congruent.

Justification: Reflections and translations are rigid transformations that preserve size and shape.

Quick Reference: Transformation Rules

💡 Key Tips for Transformations and Congruence

• ✓ Rigid transformations preserve size and shape (translations, reflections, rotations)
• ✓ Use prime notation: A → A' → A'' for sequences
• ✓ Order matters in sequences! Different orders can give different results
• ✓ Reflection over x-axis: negate y-coordinate
• ✓ Reflection over y-axis: negate x-coordinate
• ✓ Rotation 90° CCW: switch and negate first: (x, y) → (-y, x)
• ✓ Rotation 180°: negate both: (x, y) → (-x, -y)
• ✓ Congruent figures have equal corresponding parts
• ✓ CPCTC: Corresponding Parts of Congruent Triangles are Congruent
• ✓ Write congruence statements carefully - order of vertices matters!
• ✓ Always justify congruence with transformations or measurements