Infinite Geometry worksheet on graphing rotations of figures about the origin.

Here are the solutions for each rotation problem. To solve these, we apply specific rules to the coordinates $(x, y)$ of each vertex of the figure.

General Rules for Rotation about the Origin:
* $180^\circ$ Rotation: Change the sign of both coordinates: $(x, y) \rightarrow (-x, -y)$.
* $90^\circ$ Counterclockwise Rotation: Swap coordinates and change the sign of the new x-coordinate: $(x, y) \rightarrow (-y, x)$.
* $90^\circ$ Clockwise Rotation: Swap coordinates and change the sign of the new y-coordinate: $(x, y) \rightarrow (y, -x)$.

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1) Rotation $180^\circ$ about the origin

Original Points:
* $P(-2, 4)$
* $E(-5, 2)$
* $N(-3, -2)$
* $K(1, 0)$

Transformation Rule: $(x, y) \rightarrow (-x, -y)$

New Coordinates:
* $P'(-(-2), -(4)) \rightarrow P'(2, -4)$
* $E'(-(-5), -(2)) \rightarrow E'(5, -2)$
* $N'(-(-3), -(-2)) \rightarrow N'(3, 2)$
* $K'(-(1), -(0)) \rightarrow K'(-1, 0)$

Final Answer for #1: Plot points $P'(2, -4)$, $E'(5, -2)$, $N'(3, 2)$, and $K'(-1, 0)$.

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2) Rotation $180^\circ$ about the origin

Original Points:
* $V(1, 1)$
* $J(0, -1)$
* $R(3, -3)$
* $Y(4, -2)$

Transformation Rule: $(x, y) \rightarrow (-x, -y)$

New Coordinates:
* $V'(-(1), -(1)) \rightarrow V'(-1, -1)$
* $J'(-(0), -(-1)) \rightarrow J'(0, 1)$
* $R'(-(3), -(-3)) \rightarrow R'(-3, 3)$
* $Y'(-(4), -(-2)) \rightarrow Y'(-4, 2)$

Final Answer for #2: Plot points $V'(-1, -1)$, $J'(0, 1)$, $R'(-3, 3)$, and $Y'(-4, 2)$.

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3) Rotation $90^\circ$ counterclockwise about the origin

Original Points:
* $B(-4, 1)$
* $X(-3, -2)$
* $N(-5, -4)$

Transformation Rule: $(x, y) \rightarrow (-y, x)$

New Coordinates:
* $B'(-(1), -4) \rightarrow B'(-1, -4)$
* $X'(-(-2), -3) \rightarrow X'(2, -3)$
* $N'(-(-4), -5) \rightarrow N'(4, -5)$

Final Answer for #3: Plot points $B'(-1, -4)$, $X'(2, -3)$, and $N'(4, -5)$.

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4) Rotation $90^\circ$ clockwise about the origin

Original Points:
* $Y(3, 3)$
* $K(4, 3)$
* $B_x(4, 0)$ *(Note: The point on the axis is labeled $B_x$, treating as point B)*
* $U(2, -2)$

Transformation Rule: $(x, y) \rightarrow (y, -x)$

New Coordinates:
* $Y'(3, -(3)) \rightarrow Y'(3, -3)$
* $K'(3, -(4)) \rightarrow K'(3, -4)$
* $B_x'(0, -(4)) \rightarrow B_x'(0, -4)$
* $U'(-2, -(2)) \rightarrow U'(-2, -2)$

Final Answer for #4: Plot points $Y'(3, -3)$, $K'(3, -4)$, $B_x'(0, -4)$, and $U'(-2, -2)$.

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5) Rotation $90^\circ$ clockwise about the origin

Original Points:
* $J(-2, -1)$
* $V(3, 0)$
* $K(1, -3)$

Transformation Rule: $(x, y) \rightarrow (y, -x)$

New Coordinates:
* $J'(-1, -(-2)) \rightarrow J'(-1, 2)$
* $V'(0, -(3)) \rightarrow V'(0, -3)$
* $K'(-3, -(1)) \rightarrow K'(-3, -1)$

Final Answer for #5: Plot points $J'(-1, 2)$, $V'(0, -3)$, and $K'(-3, -1)$.

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6) Rotation $180^\circ$ about the origin

Original Points:
* $K(-2, -1)$
* $P(-1, -1)$
* $T(2, -2)$
* $Q(0, -4)$

Transformation Rule: $(x, y) \rightarrow (-x, -y)$

New Coordinates:
* $K'(-(-2), -(-1)) \rightarrow K'(2, 1)$
* $P'(-(-1), -(-1)) \rightarrow P'(1, 1)$
* $T'(-(2), -(-2)) \rightarrow T'(-2, 2)$
* $Q'(-(0), -(-4)) \rightarrow Q'(0, 4)$

Final Answer for #6: Plot points $K'(2, 1)$, $P'(1, 1)$, $T'(-2, 2)$, and $Q'(0, 4)$.