Rotations - Kuta Software - Free Printable

Since I can't view or access images directly, I’ll help you solve the rotation problems based on typical geometry rotation rules. Below is a step-by-step explanation of how to solve each problem, assuming standard coordinate grid setups and common shapes (like triangles or quadrilaterals) as shown in such worksheets.

We'll use the rules for rotating points about the origin:

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🔁 Rotation Rules About the Origin:

Let’s say a point has coordinates $(x, y)$.

| Rotation | New Coordinates |
|--------|-----------------|
| 90° counterclockwise | $(-y, x)$ |
| 90° clockwise | $(y, -x)$ |
| 180° | $(-x, -y)$ |
| 270° counterclockwise (or 90° clockwise) | $(y, -x)$ |

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Now let's go through each problem one by one.

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🔹 Problem 1: Rotation 180° about the origin

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

- Identify the coordinates of each vertex of the figure.
- Apply the rule: negate both $x$ and $y$.
- Plot the new points and connect them.

✔ Example:
If a vertex is at $(3, 2)$ → becomes $(-3, -2)$

> ✔ Graph the rotated image using these new points.

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🔹 Problem 2: Rotation 180° about the origin

Same rule: $(x, y) \rightarrow (-x, -y)$

- Find coordinates of triangle vertices.
- Flip signs of both coordinates.
- Plot and connect.

> ✔ Again, apply the same 180° rule.

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🔹 Problem 3: Rotation 90° counterclockwise about the origin

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

- For example:
If point is $(4, 1)$ → becomes $(-1, 4)$
- Do this for all vertices.
- Plot the new image.

> ✔ Use the transformation $(-y, x)$

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🔹 Problem 4: Rotation 90° clockwise about the origin

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

- Example:
Point $(2, 5)$ → becomes $(5, -2)$
- Apply to each vertex.
- Plot the rotated image.

> ✔ Use $(y, -x)$

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🔹 Problem 5: Rotation 90° clockwise about the origin

Same as #4: $(x, y) \rightarrow (y, -x)$

- Find original coordinates.
- Swap and negate $x$: $(x, y) \rightarrow (y, -x)$
- Plot new points.

> ✔ Same rule as above.

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🔹 Problem 6: Rotation 180° about the origin

Again: $(x, y) \rightarrow (-x, -y)$

- Negate both coordinates.
- Plot the image.

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## ✔ How to Solve Each Step-by-Step:

For each problem, follow these steps:

1. Identify the coordinates of each vertex of the original figure.
- Read from the graph (e.g., A = (2, 1), B = (4, 3), etc.)
2. Apply the correct rotation rule based on the direction and angle.
3. Plot the new coordinates on the same grid.
4. Connect the points in the same order to form the rotated image.

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## 📌 Example (Illustrative):

Suppose in Problem 1, the original shape has vertices:
- $A(1, 3)$
- $B(3, 4)$
- $C(4, 2)$
- $D(2, 1)$

After 180° rotation:
- $A' = (-1, -3)$
- $B' = (-3, -4)$
- $C' = (-4, -2)$
- $D' = (-2, -1)$

Plot these points and draw the rotated quadrilateral.

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## 🧠 Tips:
- Always label the original and image points clearly.
- Use tracing paper or careful plotting to avoid errors.
- Remember: rotations preserve size and shape — only orientation changes.

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If you can provide the coordinates of the original figures (from the image), I can compute the exact transformed coordinates for each problem.

Otherwise, just apply the rules above based on your graph!

Would you like me to walk through one specific problem with sample coordinates?