IB MYP 4-5 Maths- Rotation around a given point- Study Notes - New Syllabus
IB MYP 4-5 Maths- Rotation around a given point – Study Notes
Standard
- Rotation around a given point
IB MYP 4-5 Maths- Rotation around a given point – Study Notes – All topics
Rotation Around a Given Point
Rotation Around a Given Point
Rotation is a transformation that turns a figure around a fixed point, called the center of rotation, by a given angle and in a specified direction (clockwise or anticlockwise).
Key Features of Rotation:
Center of Rotation: The fixed point about which the shape rotates.
Angle of Rotation: The measure of the turn (e.g., \(90^\circ, 180^\circ\)).
Direction: Clockwise (CW) or Anticlockwise (ACW).
Distance: Every point of the figure remains the same distance from the center of rotation.
Shape and Size: Rotation preserves both shape and size (rigid transformation).
Common Rotation Rules about the Origin:
| Degree of Rotation | Coordinate Rules for Rotation |
|---|---|
| 90° | \( (x, y) \rightarrow (-y, x) \) |
| 180° | \( (x, y) \rightarrow (-x, -y) \) |
| 270° | \( (x, y) \rightarrow (y, -x) \) |
| -90° | \( (x, y) \rightarrow (y, -x) \) |
| -180° | \( (x, y) \rightarrow (-x, -y) \) |
| -270° | \( (x, y) \rightarrow (-y, x) \) |
Example:
Rotate point \( P(3, 2) \) about the origin through \(90^\circ\) anticlockwise.
▶️ Answer/Explanation
Step 1: For \(90^\circ\) ACW about origin, rule is: \( (x, y) \to (-y, x) \).
Step 2: Apply rule: \( (3, 2) \to (-2, 3) \).
Final Answer: New coordinates are \( (-2, 3) \).
Example:
Rotate point \( A(-4, 5) \) about the origin through \(180^\circ\).
▶️ Answer/Explanation
Step 1: For \(180^\circ\), rule is: \( (x, y) \to (-x, -y) \).
Step 2: Apply rule: \( (-4, 5) \to (4, -5) \).
Final Answer: New coordinates are \( (4, -5) \).
Example:
Rotate point \( P(6, 4) \) about point \( C(2, 2) \) by \(90^\circ\) anticlockwise.
▶️ Answer/Explanation
Step 1: Translate center to origin: Subtract \( C(2, 2) \) from \( P(6, 4) \): \( (6 – 2, 4 – 2) = (4, 2) \).
Step 2: Rotate using rule \( (x, y) \to (-y, x) \): \( (4, 2) \to (-2, 4) \).
Step 3: Translate back: Add \( C(2, 2) \): \( (-2 + 2, 4 + 2) = (0, 6) \).
Final Answer: New coordinates are \( (0, 6) \).