90° Counterclockwise: (x, y) → (-y, x)

90° Clockwise (or 270° Counterclockwise): (x, y) → (y, -x)

180° (either direction - same result): (x, y) → (-x, -y)

270° Counterclockwise (or 90° Clockwise): (x, y) → (y, -x)

360° (Full turn): (x, y) → (x, y) - back to original!

Performing Rotations

Example 1: Rotate point A(3, 2) by 90° counterclockwise around the origin.

Solution: Rule: (x, y) → (-y, x) A(3, 2) → A'(-2, 3)

Answer: A'(-2, 3)

Example 2: Rotate point B(4, -1) by 180° around the origin.

Solution: Rule: (x, y) → (-x, -y) B(4, -1) → B'(-4, 1)

Answer: B'(-4, 1)

Example 3: Rotate triangle ABC where A(2, 1), B(5, 1), C(3, 4) by 90° clockwise.

Solution: Rule for 90° clockwise: (x, y) → (y, -x)

A(2, 1) → A'(1, -2) B(5, 1) → B'(1, -5) C(3, 4) → C'(4, -3)

Answer: A'(1, -2), B'(1, -5), C'(4, -3)

Understanding Rotation Direction

Counterclockwise (CCW):

• Standard positive direction
• Like going from x-axis to y-axis
• Most common in mathematics

Clockwise (CW):

• Negative direction
• Like clock hands move
• 90° CW = 270° CCW

Tip: If not specified, assume counterclockwise!

Visualizing Rotations

90° Counterclockwise pattern:

• Quadrant I → Quadrant II
• Quadrant II → Quadrant III
• Quadrant III → Quadrant IV
• Quadrant IV → Quadrant I

180° pattern:

• Quadrant I → Quadrant III
• Quadrant II → Quadrant IV
• Quadrant III → Quadrant I
• Quadrant IV → Quadrant II

Think: Each 90° rotation moves the point to the next quadrant counterclockwise!

Dilations (Resizing)

A dilation changes the size of a shape by a scale factor.

Think of it as:

• Zooming in or out
• Enlarging or reducing a photo
• Stretching or shrinking

Key elements:

• Center of dilation - fixed point (often the origin)
• Scale factor (k) - how much to resize

Properties:

• Shape stays the same (similar figures)
• Size changes
• Distance from center multiplies by scale factor

Scale Factor (k)

k > 1: Enlargement (bigger)

• k = 2 means double the size
• k = 3 means triple the size

k = 1: No change (same size)

0 < k < 1: Reduction (smaller)

• k = 1/2 means half the size
• k = 0.5 means half the size

k < 0: Enlargement/reduction AND rotation 180°

• Rarely used in Grade 8

Dilation Rules (Center at Origin)

Rule: (x, y) → (kx, ky)

Where k is the scale factor.

Multiply both coordinates by k!

Performing Dilations

Example 1: Dilate point P(3, 4) by scale factor k = 2 with center at origin.

Solution: Rule: (x, y) → (2x, 2y) P(3, 4) → P'(6, 8)

Answer: P'(6, 8)

Distance from origin doubled!

Example 2: Dilate point Q(6, -3) by scale factor k = 1/3 with center at origin.

Solution: Rule: (x, y) → (x/3, y/3) Q(6, -3) → Q'(2, -1)

Answer: Q'(2, -1)

Distance from origin divided by 3!

Example 3: Dilate triangle ABC where A(1, 2), B(3, 2), C(2, 4) by scale factor k = 3.

Solution: Rule: (x, y) → (3x, 3y)

A(1, 2) → A'(3, 6) B(3, 2) → B'(9, 6) C(2, 4) → C'(6, 12)

Answer: A'(3, 6), B'(9, 6), C'(6, 12)

Triangle is 3 times larger!

Finding Scale Factor

Given original and image, find k:

Formula: k = (image coordinate)/(original coordinate)

Example: Point A(4, 6) dilates to A'(2, 3). Find scale factor.

Solution: k = 2/4 = 1/2 (using x-coordinates) or k = 3/6 = 1/2 (using y-coordinates)

Answer: k = 1/2 (reduction to half size)

Properties of Dilations

What stays the same:

• Shape - angles stay equal
• Ratios - proportions preserved
• Parallelism - parallel lines stay parallel

What changes:

• Size - lengths multiply by |k|
• Perimeter - multiplies by |k|
• Area - multiplies by k²

Example: Triangle with area 10 cm² dilated by k = 3 New area = 10 × 3² = 10 × 9 = 90 cm²

Combining Rotations and Dilations

Example: Point A(2, 3) is rotated 90° counterclockwise, then dilated by k = 2. Find final position.

Solution:

Step 1: Rotation (x, y) → (-y, x) A(2, 3) → (-3, 2)

Step 2: Dilation (x, y) → (2x, 2y) (-3, 2) → (-6, 4)

Answer: Final position (-6, 4)

Note: Order matters! Different order = different result!

Rigid vs. Non-Rigid Transformations

Rigid Transformations (Isometries):

• Preserve size AND shape
• Figures are CONGRUENT
• Examples: Translation, Reflection, Rotation

Non-Rigid Transformations:

• Preserve shape but NOT size
• Figures are SIMILAR (not congruent)
• Example: Dilation

Real-World Applications

Rotations:

Clock hands: Rotating around center

• Hour hand: 360° in 12 hours = 30°/hour
• Minute hand: 360° in 60 minutes = 6°/minute

Wheels and gears: Spinning machinery

Ferris wheels: Rotating around axis

Dance/gymnastics: Spinning moves

Navigation: Compass directions

Dilations:

Photography: Zoom in/out

• Digital zoom = dilation with k > 1
• Zoom out = dilation with k < 1

Maps: Scale drawings

• 1 inch = 10 miles means k = 1/(10×63360)

Architecture: Scale models

• 1:100 scale means k = 1/100

Computer graphics: Resizing images

Photocopiers: Enlarging/reducing documents

Rotation Symmetry

Some shapes look the same after rotation!

Rotation symmetry: Shape looks unchanged after rotating less than 360°

Order of symmetry: How many times it looks the same in one full turn

Examples:

• Square: 90° rotation looks same (order 4)
• Equilateral triangle: 120° rotation looks same (order 3)
• Regular hexagon: 60° rotation looks same (order 6)
• Circle: Any rotation looks same (infinite order)

Common Mistakes to Avoid

❌ Mistake 1: Mixing up rotation rules

• Wrong: 90° CCW is (y, x)
• Right: 90° CCW is (-y, x)

❌ Mistake 2: Forgetting negative signs in rotations

• 180° rotation changes BOTH signs!
• (3, 4) → (-3, -4), not (3, 4)

❌ Mistake 3: Not multiplying BOTH coordinates in dilation

• Wrong: (2, 3) with k=2 → (4, 3)
• Right: (2, 3) with k=2 → (4, 6)

❌ Mistake 4: Confusing enlargement and reduction

• k > 1 makes it BIGGER
• 0 < k < 1 makes it SMALLER

❌ Mistake 5: Wrong order in combined transformations

• Order matters! Rotation then dilation ≠ dilation then rotation

Problem-Solving Strategy

For Rotations:

1. Identify center and angle
2. Determine direction (CW or CCW)
3. Apply correct rule
4. Plot new points

For Dilations:

1. Identify center and scale factor
2. Determine if enlargement or reduction
3. Multiply coordinates by k
4. Plot new points

For Combined:

1. Do transformations in order given
2. Use result from first as input to second
3. Check final answer

Quick Reference

Rotation Rules (around origin):

• 90° CCW: (x, y) → (-y, x)
• 90° CW: (x, y) → (y, -x)
• 180°: (x, y) → (-x, -y)
• 270° CCW: (x, y) → (y, -x)

Dilation Rule (center at origin):

• (x, y) → (kx, ky)
• k > 1: enlargement
• 0 < k < 1: reduction
• k = 1: no change

Properties:

• Rotations: preserve size and shape (congruent)
• Dilations: preserve shape only (similar)

Practice Tips

Tip 1: Draw it!

• Sketch before calculating
• Visualize the transformation
• Check if answer makes sense

Tip 2: Use reference points

• Origin (0,0) never moves in rotations/dilations centered there
• Points on axes are easier to track

Tip 3: Check with distance formula

• After rotation, distance from center should be same
• After dilation, distance should multiply by k

Tip 4: Remember the sign patterns

• Each rotation rule has specific sign pattern
• Practice until automatic!

Summary

Rotations turn shapes around a point:

• Common angles: 90°, 180°, 270°
• Direction: clockwise or counterclockwise
• Preserve size and shape (congruent)
• Rules depend on angle and direction

Dilations resize shapes from a center:

• Scale factor k determines new size
• k > 1: enlargement
• 0 < k < 1: reduction
• Preserve shape, not size (similar)
• Rule: multiply coordinates by k

Both are essential transformations:

• Rotations for spinning and turning
• Dilations for scaling and resizing
• Together: complete our transformation toolkit!

Understanding rotations and dilations completes the study of geometric transformations and connects to similarity, congruence, and real-world applications!