Math worksheet on geometric rotations, featuring six exercises with shapes to rotate on coordinate planes.

Problem Description:

The task involves performing rotations of geometric shapes about the origin on a coordinate grid. The rotations are specified as either 180° or 90° (either clockwise or counterclockwise). We need to solve each rotation step by step.

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Key Concepts:

1. Rotation Rules:
- 180° Rotation About the Origin:
For any point \((x, y)\), the new coordinates after a 180° rotation are \((-x, -y)\).
- 90° Counterclockwise (ccw) Rotation About the Origin:
For any point \((x, y)\), the new coordinates after a 90° ccw rotation are \((-y, x)\).
- 90° Clockwise Rotation About the Origin:
For any point \((x, y)\), the new coordinates after a 90° clockwise rotation are \((y, -x)\).

2. Steps:
- Identify the original coordinates of the vertices of the shape.
- Apply the appropriate rotation rule to each vertex.
- Plot the new coordinates on the grid to form the rotated shape.

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Solution for Each Problem:

#### 1) Rotation: 180° about the origin
- Original Shape: The shape is a right triangle with vertices at \((1, 2)\), \((1, 3)\), and \((2, 2)\).
- Rotation Rule: \((x, y) \rightarrow (-x, -y)\).
- New Coordinates:
- \((1, 2) \rightarrow (-1, -2)\)
- \((1, 3) \rightarrow (-1, -3)\)
- \((2, 2) \rightarrow (-2, -2)\)
- Result: Plot the new points \((-1, -2)\), \((-1, -3)\), and \((-2, -2)\) to form the rotated triangle.

#### 2) Rotation: 180° about the origin
- Original Shape: The shape is a quadrilateral with vertices at \((1, 2)\), \((2, 2)\), \((2, 3)\), and \((1, 3)\).
- Rotation Rule: \((x, y) \rightarrow (-x, -y)\).
- New Coordinates:
- \((1, 2) \rightarrow (-1, -2)\)
- \((2, 2) \rightarrow (-2, -2)\)
- \((2, 3) \rightarrow (-2, -3)\)
- \((1, 3) \rightarrow (-1, -3)\)
- Result: Plot the new points \((-1, -2)\), \((-2, -2)\), \((-2, -3)\), and \((-1, -3)\) to form the rotated quadrilateral.

#### 3) Rotation: 90° ccw about the origin
- Original Shape: The shape is a quadrilateral with vertices at \((1, 2)\), \((2, 2)\), \((2, 3)\), and \((1, 3)\).
- Rotation Rule: \((x, y) \rightarrow (-y, x)\).
- New Coordinates:
- \((1, 2) \rightarrow (-2, 1)\)
- \((2, 2) \rightarrow (-2, 2)\)
- \((2, 3) \rightarrow (-3, 2)\)
- \((1, 3) \rightarrow (-3, 1)\)
- Result: Plot the new points \((-2, 1)\), \((-2, 2)\), \((-3, 2)\), and \((-3, 1)\) to form the rotated quadrilateral.

#### 4) Rotation: 180° about the origin
- Original Shape: The shape is a quadrilateral with vertices at \((1, 2)\), \((2, 2)\), \((2, 3)\), and \((1, 3)\).
- Rotation Rule: \((x, y) \rightarrow (-x, -y)\).
- New Coordinates:
- \((1, 2) \rightarrow (-1, -2)\)
- \((2, 2) \rightarrow (-2, -2)\)
- \((2, 3) \rightarrow (-2, -3)\)
- \((1, 3) \rightarrow (-1, -3)\)
- Result: Plot the new points \((-1, -2)\), \((-2, -2)\), \((-2, -3)\), and \((-1, -3)\) to form the rotated quadrilateral.

#### 5) Rotation: 90° clockwise about the origin
- Original Shape: The shape is a quadrilateral with vertices at \((1, 2)\), \((2, 2)\), \((2, 3)\), and \((1, 3)\).
- Rotation Rule: \((x, y) \rightarrow (y, -x)\).
- New Coordinates:
- \((1, 2) \rightarrow (2, -1)\)
- \((2, 2) \rightarrow (2, -2)\)
- \((2, 3) \rightarrow (3, -2)\)
- \((1, 3) \rightarrow (3, -1)\)
- Result: Plot the new points \((2, -1)\), \((2, -2)\), \((3, -2)\), and \((3, -1)\) to form the rotated quadrilateral.

#### 6) Rotation: 90° ccw about the origin
- Original Shape: The shape is a quadrilateral with vertices at \((1, 2)\), \((2, 2)\), \((2, 3)\), and \((1, 3)\).
- Rotation Rule: \((x, y) \rightarrow (-y, x)\).
- New Coordinates:
- \((1, 2) \rightarrow (-2, 1)\)
- \((2, 2) \rightarrow (-2, 2)\)
- \((2, 3) \rightarrow (-3, 2)\)
- \((1, 3) \rightarrow (-3, 1)\)
- Result: Plot the new points \((-2, 1)\), \((-2, 2)\), \((-3, 2)\), and \((-3, 1)\) to form the rotated quadrilateral.

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Final Answer:

Each problem requires plotting the new coordinates based on the rotation rules. The solutions are summarized as follows:

1. 180° Rotation: New points are \((-1, -2)\), \((-1, -3)\), and \((-2, -2)\).
2. 180° Rotation: New points are \((-1, -2)\), \((-2, -2)\), \((-2, -3)\), and \((-1, -3)\).
3. 90° ccw Rotation: New points are \((-2, 1)\), \((-2, 2)\), \((-3, 2)\), and \((-3, 1)\).
4. 180° Rotation: New points are \((-1, -2)\), \((-2, -2)\), \((-2, -3)\), and \((-1, -3)\).
5. 90° Clockwise Rotation: New points are \((2, -1)\), \((2, -2)\), \((3, -2)\), and \((3, -1)\).
6. 90° ccw Rotation: New points are \((-2, 1)\), \((-2, 2)\), \((-3, 2)\), and \((-3, 1)\).

\[
\boxed{\text{See detailed steps above for each rotation.}}
\]