Geometry Worksheets | Transformations Worksheets - Free Printable

Since I can't view or access images directly, I can’t see the specific shapes or their coordinates in your uploaded image. However, I can guide you through how to solve each rotation problem based on standard coordinate geometry rules for rotations about the origin.

Let’s go through each of the six problems step-by-step and explain how to solve them. You can then apply these rules to the figures in your image.

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🔁 General Rules for Rotations About the Origin

For a point $(x, y)$:

1. Rotation 90° counterclockwise (ccw):
$$(x, y) \rightarrow (-y, x)$$

2. Rotation 90° clockwise:
$$(x, y) \rightarrow (y, -x)$$

3. Rotation 180°:
$$(x, y) \rightarrow (-x, -y)$$

4. Rotation 270° ccw (same as 90° clockwise):
$$(x, y) \rightarrow (y, -x)$$

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Now let's go through each problem.

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1) Rotation: 180° about the origin

- For every point $(x, y)$, apply: $(-x, -y)$
- Example: If a vertex is at $(2, 1)$, it becomes $(-2, -1)$
- Rotate all vertices of the shape using this rule.
- Then draw the new shape on the grid.

✔ Tip: The rotated figure will be upside-down and mirrored across the origin.

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2) Rotation: 180° about the origin

- Same rule: $(x, y) \rightarrow (-x, -y)$
- Apply to each vertex of the green shape.
- This is a full flip through the origin.

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3) Rotation: 90° ccw about the origin

- Use rule: $(x, y) \rightarrow (-y, x)$
- Example: A point at $(2, 3)$ → $(-3, 2)$
- Rotate each vertex accordingly.
- The shape will appear "turned left" by a quarter turn.

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4) Rotation: 180° about the origin

- Again: $(x, y) \rightarrow (-x, -y)$
- All points are reflected through the origin.

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5) Rotation: 90° clockwise about the origin

- Rule: $(x, y) \rightarrow (y, -x)$
- Example: $(2, 3)$ → $(3, -2)$
- The shape turns right by 90 degrees.

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6) Rotation: 90° ccw about the origin

- Rule: $(x, y) \rightarrow (-y, x)$
- Same as #3 — just apply to the L-shaped figure.

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✔ How to Solve Each Problem Step-by-Step:

1. Identify the coordinates of each vertex of the original shape.
2. Apply the correct rotation rule based on the problem.
3. Plot the new coordinates on the same grid.
4. Connect the points to form the rotated image.

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📌 Example Walkthrough (for problem 1):

Suppose the shape has vertices at:
- $A(1, 2)$
- $B(2, 2)$
- $C(2, 1)$

After 180° rotation:
- $A' = (-1, -2)$
- $B' = (-2, -2)$
- $C' = (-2, -1)$

Plot these new points and connect them.

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🔍 Final Tips:

- Always rotate each vertex, not the whole shape.
- Use graph paper carefully.
- Check if the orientation makes sense: 90° ccw should look like turning the page left; clockwise like turning right.
- After 180°, the shape should face opposite direction and be centered at origin.

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If you can describe one of the shapes (e.g., list the coordinates of its vertices), I can help you compute the rotated image exactly.

Otherwise, use the rules above to complete each problem on your worksheet.

Let me know if you'd like a printable version of the solution steps!