Rotations on the Coordinate Plane | Worksheet - Free Printable

Since I can't view or access images directly, I can’t see the specific details of your image. However, based on the text you provided, it appears to be a geometry worksheet about rotations on the coordinate plane, specifically asking you to rotate triangles around the origin.

Let me walk you through how to solve these types of problems and explain the process step by step. If you provide the coordinates of the original triangle (e.g., ΔABC), I can help you find the rotated image.

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🔷 Problem Overview:

You are asked to graph the image of a triangle after a 90° counterclockwise rotation around the origin (0, 0).

There are two common rotations around the origin:

#### ✔ 90° Counterclockwise Rotation Rule:
For any point $(x, y)$, the image after a 90° counterclockwise rotation is:
$$
(x, y) \rightarrow (-y, x)
$$

#### ✔ 180° Rotation Rule (if needed):
$$
(x, y) \rightarrow (-x, -y)
$$

#### ✔ 270° Counterclockwise (or 90° Clockwise):
$$
(x, y) \rightarrow (y, -x)
$$

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📌 Step-by-Step Solution (Example):

Suppose triangle $ \triangle ABC $ has vertices at:
- $ A(2, 1) $
- $ B(4, 3) $
- $ C(1, 4) $

We want to rotate this triangle 90° counterclockwise about the origin.

Apply the rule: $ (x, y) \rightarrow (-y, x) $

- $ A(2, 1) \rightarrow (-1, 2) $
- $ B(4, 3) \rightarrow (-3, 4) $
- $ C(1, 4) \rightarrow (-4, 1) $

So the new triangle $ \triangle A'B'C' $ has vertices:
- $ A'(-1, 2) $
- $ B'(-3, 4) $
- $ C'(-4, 1) $

Now plot these points on the coordinate plane and connect them to form the rotated triangle.

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🎯 How to Graph It:

1. Draw the original triangle using the given coordinates.
2. Apply the rotation rule to each vertex.
3. Plot the new points.
4. Connect the new points to form the rotated image.
5. Label the image (e.g., $ \triangle A'B'C' $).

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🔍 Tips:

- The shape and size remain the same — only orientation changes.
- Rotations preserve distances and angles.
- The origin stays fixed during rotation.

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❓If You Provide Coordinates:

Please tell me the coordinates of the triangle in your image (e.g., "Triangle ABC has points A(1,2), B(3,1), C(2,4)") and specify which rotation you need (90°, 180°, etc.), and I’ll give you the exact solution with steps.

Alternatively, if you describe what’s in the image (like the positions of the points), I can guide you through solving it.

Let me know!