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Learning Resources

Rotation

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 10 Pathway: N/A

First Principles

Objective: Master the transformation of rotating 2D points and shapes about the origin or any arbitrary centre point by a specified angle and direction.

Intuition: Imagine the Lake Turkana wind turbines. As a blade spins around its central hub, the blade length never stretches or shrinks, and the hub stays completely fixed. Every single point on the blade sweeps out an arc of a circle. The fixed hub is the centre of rotation, and the degree of turn is the angle of rotation.

1. Core Properties of a Rotation

  • Isometry (Rigid Motion): Distances and angles are preserved. The image is congruent to the object.
  • Centre of Rotation: The unique invariant point that does not change position during the turn.
  • Direction: By standard convention in mathematics:
    • Anticlockwise (Counter-Clockwise): Positive angle (\(+\theta\)).
    • Clockwise: Negative angle (\(-\theta\)).

2. Geometric Breakdown

For any point \(P(x,y)\) at distance \(r\) and angle \(\alpha\) from the origin: \[ x = r\cos\alpha, \quad y = r\sin\alpha \] Rotating by angle \(\theta\) produces a new angle \(\alpha + \theta\): \[ x' = r\cos(\alpha + \theta) = r(\cos\alpha\cos\theta - \sin\alpha\sin\theta) = x\cos\theta - y\sin\theta \] \[ y' = r\sin(\alpha + \theta) = r(\sin\alpha\cos\theta + \cos\alpha\sin\theta) = x\sin\theta + y\cos\theta \]

Key Formulas

1. Standard Rotations About the Origin \((0,0)\):

  • \(90^{\circ}\) Anticlockwise (or \(270^{\circ}\) Clockwise): \[ (x, y) \mapsto (-y, x) \]
  • \(180^{\circ}\) (Clockwise or Anticlockwise): \[ (x, y) \mapsto (-x, -y) \]
  • \(270^{\circ}\) Anticlockwise (or \(90^{\circ}\) Clockwise): \[ (x, y) \mapsto (y, -x) \]
  • \(360^{\circ}\) Full Turn: \[ (x, y) \mapsto (x, y) \]

2. General Rotation Formula About Origin by Angle \(\theta\):

\[ x' = x\cos\theta - y\sin\theta \] \[ y' = x\sin\theta + y\cos\theta \]

Note: Use \(+\theta\) for anticlockwise and \(-\theta\) for clockwise.

3. Rotation About Any Arbitrary Centre \((h, k)\):

  1. Translate centre to origin: \(x_{\text{rel}} = x - h\), \(y_{\text{rel}} = y - k\).
  2. Apply rotation to the relative coordinates.
  3. Translate back: add \(h\) to \(x'\) and \(k\) to \(y'\). \[ x' = h + (x-h)\cos\theta - (y-k)\sin\theta \] \[ y' = k + (x-h)\sin\theta + (y-k)\cos\theta \]

Worked Examples

Problem 1 (Easy): Rotate the point \(A(5, -2)\) by \(90^{\circ}\) anticlockwise about the origin. Determine the coordinates of the image point \(A'\).
  1. Recall the transformation rule: For a \(90^{\circ}\) anticlockwise rotation about \((0,0)\), the rule is \((x, y) \mapsto (-y, x)\).
  2. Substitute coordinates: Here \(x = 5\) and \(y = -2\).
  3. Calculate image: \[ x' = -(-2) = 2 \] \[ y' = 5 \]
  4. Result: \(A'(2, 5)\).
Problem 2 (Medium): A landmark beacon at \(P(4, 7)\) is rotated by \(180^{\circ}\) about the origin. A second transformation then rotates the image by \(90^{\circ}\) clockwise about the origin. Find the final coordinates.
  1. Step 1 (First rotation of \(180^{\circ}\)): The rule is \((x, y) \mapsto (-x, -y)\). \[ P(4, 7) \mapsto P_1(-4, -7) \]
  2. Step 2 (Second rotation of \(90^{\circ}\) clockwise): Rotating \(90^{\circ}\) clockwise is equivalent to \(270^{\circ}\) anticlockwise, with mapping \((x, y) \mapsto (y, -x)\). \[ x_2 = y_1 = -7 \] \[ y_2 = -x_1 = -(-4) = 4 \]
  3. Combined check: Total rotation is \(180^{\circ} - 90^{\circ} = +90^{\circ}\) anticlockwise: \((4,7) \mapsto (-7, 4)\). Both methods match.
  4. Result: \((-7, 4)\).
Problem 3 (Hard): Rotate the vertex \(B(5, 6)\) by \(90^{\circ}\) anticlockwise about the centre \(C(2, 2)\). Find the image point \(B'\).
  1. Shift relative to centre \(C(h,k) = (2,2)\): \[ x_{\text{rel}} = x - h = 5 - 2 = 3 \] \[ y_{\text{rel}} = y - k = 6 - 2 = 4 \]
  2. Apply \(90^{\circ}\) anticlockwise rotation to relative point: \[ x'_{\text{rel}} = -y_{\text{rel}} = -4 \] \[ y'_{\text{rel}} = x_{\text{rel}} = 3 \]
  3. Shift back to actual plane: \[ x' = h + x'_{\text{rel}} = 2 + (-4) = -2 \] \[ y' = k + y'_{\text{rel}} = 2 + 3 = 5 \]
  4. Result: \(B'(-2, 5)\).

Common Mistakes

Mistake 1: Confusing Clockwise and Anticlockwise signs.
Correction In standard Cartesian mathematics, positive angles turn anticlockwise, while negative angles turn clockwise. If a question asks for a \(90^{\circ}\) clockwise rotation, do not use \((-y, x)\); instead use \((y, -x)\).
Why it feels right Clocks turn clockwise in daily life, leading learners to assume clockwise is positive. Always double check standard mathematical orientation.
Mistake 2: Forgetting to translate when rotating about a centre \((h,k) \neq (0,0)\).
Correction Directly applying \((-y, x)\) to \((5, 6)\) gives \((-6, 5)\), which rotates around the origin \((0,0)\), NOT around the pivot \((2,2)\). You must shift relative to the pivot first, rotate, and then shift back.
Mistake 3: Believing that distance or shape changes.
Correction Rotation is a rigid transformation (isometry). Side lengths, angles, and area are completely invariant. Only the position and orientation change.

Real World

Lake Turkana Wind Power: Engineers calculate the coordinates of turbine blade tips rotating around the nacelle hub to optimize aerodynamic clearance and wind capture.
Robotics and Computer Vision in Kenyan Agtech: Robotic arms sorting tea leaves or avocados rotate their grippers around specific joints using transformation matrices.
Traffic Roundabouts (e.g., Globe Cinema Roundabout, Nairobi): Vehicles rotate around the central island at continuous angles before exiting onto arterial roads.
Satellite Navigation and Solar Tracking: Solar arrays in Garissa rotate continuously throughout the day to remain perpendicular to incoming sunlight.

Practice

A point \(P(4, 7)\) is rotated by \(90^{\circ}\) anticlockwise about the origin \((0,0)\). What is the resulting y-coordinate of the image point? (Type only the number, e.g., 42)
Review the concepts above.
A solar panel in Garissa is rotated 65° clockwise in the morning, and later rotated an additional 55° clockwise in the afternoon. What is the total clockwise rotation of the panel in degrees? (Type only the number, e.g., 42)
Review the concepts above.
A survey marker at point \(Q(3, -5)\) is rotated by \(180^{\circ}\) about the origin \((0,0)\). What is the resulting x-coordinate of the marker? (Type only the number, e.g., 42)
Review the concepts above.
A textile designer in Nairobi rotates a kanga pattern piece by 135° clockwise about a central pin. What is the equivalent anticlockwise angle of rotation in degrees that reaches the exact same position? (Type only the number, e.g., 42)
Review the concepts above.
A drone marker at point \(A(6, 4)\) is rotated \(90^{\circ}\) anticlockwise about the pivot point \(C(2, 1)\). What is the resulting x-coordinate of the marker? (Type only the number, e.g., 42)
Review the concepts above.
A wind turbine blade tip at point \(B(5, 7)\) is rotated \(90^{\circ}\) clockwise about the hub located at \(H(1, 3)\). What is the resulting y-coordinate of the blade tip? (Type only the number, e.g., 42)
Review the concepts above.