Math worksheet for practicing rotations of geometric figures on a coordinate plane.

To solve these rotation problems, we need to apply specific rules for rotating points $(x, y)$ on a coordinate plane around the origin $(0,0)$.

Here are the rules we will use:
1. Rotation $180^\circ$: The rule is $(x, y) \rightarrow (-x, -y)$. This means you flip the sign of both coordinates. It looks like the shape has been turned upside down and moved to the opposite side.
2. Rotation $90^\circ$ counterclockwise (ccw): The rule is $(x, y) \rightarrow (-y, x)$. You swap the numbers and change the sign of the first one (the new $x$).
3. Rotation $90^\circ$ clockwise: The rule is $(x, y) \rightarrow (y, -x)$. You swap the numbers and change the sign of the second one (the new $y$).

Let's solve each problem step-by-step by identifying the vertices (corners) of the original green shape and applying the rules.

Problem 1) Rotation: $180^\circ$ about the origin
* Original Vertices: Looking at the grid, the triangle has corners at $(-1, 1)$, $(-1, 2)$, and $(1, 2)$.
* Apply Rule $(-x, -y)$:
* $(-1, 1)$ becomes $(1, -1)$
* $(-1, 2)$ becomes $(1, -2)$
* $(1, 2)$ becomes $(-1, -2)$
* Result: Draw a triangle connecting $(1, -1)$, $(1, -2)$, and $(-1, -2)$.

Problem 2) Rotation: $180^\circ$ about the origin
* Original Vertices: The quadrilateral has corners at $(1, 4)$, $(3, 4)$, $(3, 2)$, and $(1, 2)$.
* Apply Rule $(-x, -y)$:
* $(1, 4)$ becomes $(-1, -4)$
* $(3, 4)$ becomes $(-3, -4)$
* $(3, 2)$ becomes $(-3, -2)$
* $(1, 2)$ becomes $(-1, -2)$
* Result: Draw the shape connecting $(-1, -4)$, $(-3, -4)$, $(-3, -2)$, and $(-1, -2)$.

Problem 3) Rotation: $90^\circ$ ccw about the origin
* Original Vertices: The triangle has corners at $(2, 5)$, $(2, 3)$, and $(4, 3)$.
* Apply Rule $(-y, x)$:
* $(2, 5)$ becomes $(-5, 2)$
* $(2, 3)$ becomes $(-3, 2)$
* $(4, 3)$ becomes $(-3, 4)$
* Result: Draw a triangle connecting $(-5, 2)$, $(-3, 2)$, and $(-3, 4)$.

Problem 4) Rotation: $180^\circ$ about the origin
* Original Vertices: The L-shape has outer corners at $(1, 4)$, $(3, 4)$, $(3, 1)$, $(2, 1)$, $(2, 2)$, and $(1, 2)$. Let's just track the main "L" structure.
* Top-right corner $(3, 4)$ becomes $(-3, -4)$.
* Bottom-inner corner $(2, 2)$ becomes $(-2, -2)$.
* Top-left corner $(1, 4)$ becomes $(-1, -4)$.
* Bottom-left corner $(1, 1)$ [implied vertex] becomes $(-1, -1)$.
* Actually, let's look closer at the vertices: $(1,4), (3,4), (3,1), (2,1), (2,2), (1,2)$.
* Rotated: $(-1,-4), (-3,-4), (-3,-1), (-2,-1), (-2,-2), (-1,-2)$.
* Result: Draw the rotated L-shape in the third quadrant using these points.

Problem 5) Rotation: $90^\circ$ clockwise about the origin
* Original Vertices: The stepped shape has vertices at $(-4, 1), (-4, 2), (-3, 2), (-3, 3), (-2, 3), (-2, 4), (-1, 4), (-1, 1)$.
* Apply Rule $(y, -x)$:
* $(-4, 1) \rightarrow (1, 4)$
* $(-4, 2) \rightarrow (2, 4)$
* $(-3, 2) \rightarrow (2, 3)$
* $(-3, 3) \rightarrow (3, 3)$
* $(-2, 3) \rightarrow (3, 2)$
* $(-2, 4) \rightarrow (4, 2)$
* $(-1, 4) \rightarrow (4, 1)$
* $(-1, 1) \rightarrow (1, 1)$
* Result: Draw the shape in the first quadrant connecting these new points.

Problem 6) Rotation: $90^\circ$ ccw about the origin
* Original Vertices: The complex shape has vertices including $(2, 5), (4, 5), (4, 4), (5, 4), (5, 2), (3, 2), (3, 3), (2, 3)$.
* Apply Rule $(-y, x)$:
* $(2, 5) \rightarrow (-5, 2)$
* $(4, 5) \rightarrow (-5, 4)$
* $(4, 4) \rightarrow (-4, 4)$
* $(5, 4) \rightarrow (-4, 5)$
* $(5, 2) \rightarrow (-2, 5)$
* $(3, 2) \rightarrow (-2, 3)$
* $(3, 3) \rightarrow (-3, 3)$
* $(2, 3) \rightarrow (-3, 2)$
* Result: Draw the shape in the second quadrant using these coordinates.

Final Answer:
1) Plot triangle with vertices: (1, -1), (1, -2), (-1, -2)
2) Plot quadrilateral with vertices: (-1, -4), (-3, -4), (-3, -2), (-1, -2)
3) Plot triangle with vertices: (-5, 2), (-3, 2), (-3, 4)
4) Plot L-shape with vertices: (-1, -4), (-3, -4), (-3, -1), (-2, -1), (-2, -2), (-1, -2)
5) Plot stepped shape with vertices: (1, 4), (2, 4), (2, 3), (3, 3), (3, 2), (4, 2), (4, 1), (1, 1)
6) Plot complex shape with vertices: (-5, 2), (-5, 4), (-4, 4), (-4, 5), (-2, 5), (-2, 3), (-3, 3), (-3, 2)