# Rotations and Dilations Transformations continue with rotations (turns) and dilations (resizing)! While translations and reflections move shapes, rotations spin them and dilations change their size. These transformations are essential for geometry, art, and computer graphics. --- ## Rotations (Turns) A **rotation** turns a shape around a fixed point called the center of rotation. **Think of it as:** - Spinning a shape - Turning a clock hand - Rotating a wheel **Key elements:** - **Center of rotation** - fixed point (often the origin) - **Angle of rotation** - how far to turn (in degrees) - **Direction** - clockwise or counterclockwise **Properties:** - Shape and size stay the same - Distance from center stays the same - Orientation changes (shape turns) --- ## Rotation Rules (Around Origin) **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!