Rotation is a transformation that turns a geometric shape around a fixed point. Explore rotation rules with LaTeX notation, visualize transformations of points and triangles around the origin, and practice solving rotation problems.
What is Rotation?
Rotation is a transformation that turns a shape around a fixed point called the center of rotation through a specific angle and direction.
- Center of Rotation: The fixed point (often the origin \((0,0)\)).
- Angle of Rotation: How far to rotate (e.g., \(90^\circ\), \(180^\circ\)).
- Direction: Clockwise (CW) or Counterclockwise (CCW).
Common Rotation Rules (Around the Origin)
Let \((x,y)\) be the coordinates of a point. The coordinates \((x',y')\) after rotation around the origin are given by:
| Rotation | Coordinate Rule | Matrix Form |
|---|---|---|
| \(90^\circ\) CCW | \((x, y) \rightarrow (x' = -y, y' = x)\) | \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\) |
| \(180^\circ\) | \((x, y) \rightarrow (x' = -x, y' = -y)\) | \(\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\) |
| \(270^\circ\) CCW | \((x, y) \rightarrow (x' = y, y' = -x)\) | \(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\) |
| \(360^\circ\) | \((x, y) \rightarrow (x' = x, y' = y)\) | \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\) |
For a general rotation by angle \(\theta\):
\[\begin{aligned} x' &= x \cos\theta - y \sin\theta \\ y' &= x \sin\theta + y \cos\theta \end{aligned}\]Interactive Activity 1: Rotate Your Own Points
Click on the canvas to add points. Use the slider or angle input to rotate them. Points are labeled A, B, C (red) and their rotated images A', B', C' (blue).
Interactive Activity 2: Rotate a Triangle
Visualize the rotation of a triangle around the origin using the interactive controls below.
Practice Problems & Solutions
Problem 1: Point rotation by 90 degrees CCW
Question: Point \(P(3, 4)\) is rotated \(90^\circ\) counterclockwise about the origin. Find the coordinates of \(P'\).
Solution:
Using the \(90^\circ\) CCW rule: \((x, y) \rightarrow (-y, x)\)
\[\begin{aligned} P(3, 4) &\rightarrow P'(-4, 3) \\ x' &= -y = -4 \\ y' &= x = 3 \end{aligned}\]Answer: \(P'(-4, 3)\)
Problem 2: Triangle vertices rotation by 180 degrees
Question: Triangle \(ABC\) has vertices \(A(1, 1)\), \(B(4, 1)\), and \(C(2, 5)\). Find the vertices after a \(180^\circ\) rotation about the origin.
Solution:
Using the \(180^\circ\) rotation rule: \((x, y) \rightarrow (-x, -y)\)
\[\begin{aligned} A(1, 1) &\rightarrow A'(-1, -1) \\ B(4, 1) &\rightarrow B'(-4, -1) \\ C(2, 5) &\rightarrow C'(-2, -5) \end{aligned}\]Answer: \(A'(-1, -1)\), \(B'(-4, -1)\), \(C'(-2, -5)\)
Problem 3: Point rotation by 270 degrees CCW
Question: Point \(Q(-2, 5)\) is rotated \(270^\circ\) counterclockwise about the origin. Find the coordinates of \(Q'\).
Solution:
Using the \(270^\circ\) CCW rule: \((x, y) \rightarrow (y, -x)\)
\[\begin{aligned} Q(-2, 5) &\rightarrow Q'(5, 2) \\ x' &= y = 5 \\ y' &= -x = -(-2) = 2 \end{aligned}\]Answer: \(Q'(5, 2)\)