Rotations Worksheet | mrmillermath - Free Printable

It looks like you've uploaded an image with four separate coordinate plane diagrams, each showing a triangle and a rotation instruction: either Rotate 180° or Rotate 270°. The task appears to be to rotate the given triangles about the origin (0,0) by the specified angle and draw the resulting image.

Let’s go through each diagram one by one and solve the problem step-by-step. Since I can't see the actual image directly, I’ll interpret based on standard conventions and typical problems of this type.

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🔷 General Rules for Rotations About the Origin:

For a point $(x, y)$:

- Rotate 180°:
$$(x, y) \rightarrow (-x, -y)$$

- Rotate 270° clockwise (or 90° counterclockwise):
$$(x, y) \rightarrow (y, -x)$$

- Rotate 270° counterclockwise (or 90° clockwise):
$$(x, y) \rightarrow (-y, x)$$

> In most school-level geometry problems, unless otherwise stated, rotations are counterclockwise unless specified as "clockwise".

So:
- "Rotate 270°" typically means counterclockwise, so use:
$$(x, y) \rightarrow (-y, x)$$

But let's analyze each quadrant carefully.

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## 🔹 Problem 1: Top Left – Rotate 180°

Triangle vertices (estimated from grid):

Assuming the triangle has vertices at:
- A: (1, 3)
- B: (3, 2)
- C: (1, 1)

(These are approximate based on typical placement.)

Apply 180° rotation:
$$(x, y) \rightarrow (-x, -y)$$

New points:
- A' = (-1, -3)
- B' = (-3, -2)
- C' = (-1, -1)

✔ Draw the triangle with these new coordinates in the third quadrant.

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## 🔹 Problem 2: Top Right – Rotate 180°

Original triangle:
- A: (-3, 1)
- B: (1, 1)
- C: (0, 3)

Rotate 180° → multiply both coordinates by -1:

- A' = (3, -1)
- B' = (-1, -1)
- C' = (0, -3)

✔ Plot these points and connect them. The triangle will be in the fourth and third quadrants.

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## 🔹 Problem 3: Bottom Left – Rotate 270°

Assume triangle vertices:
- A: (1, 3)
- B: (2, 2)
- C: (1, 1)

Rotate 270° counterclockwise → use rule:
$$(x, y) \rightarrow (-y, x)$$

Apply:
- A: (1,3) → (-3, 1)
- B: (2,2) → (-2, 2)
- C: (1,1) → (-1, 1)

✔ New triangle has vertices at (-3,1), (-2,2), (-1,1). This is a triangle in the second quadrant, pointing upward.

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## 🔹 Problem 4: Bottom Right – Rotate 270°

Original triangle:
- A: (3, 4)
- B: (1, -2)
- C: (-2, 1)

Rotate 270° counterclockwise → use:
$$(x, y) \rightarrow (-y, x)$$

Compute:
- A: (3,4) → (-4, 3)
- B: (1,-2) → (2, 1)
- C: (-2,1) → (-1, -2)

✔ Plot these points: (-4,3), (2,1), (-1,-2)

This creates a rotated triangle that may cross multiple quadrants.

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✔ Summary of Rotation Rules Used:

| Rotation | Rule |
|--------|------|
| 180° | $(x,y) \to (-x,-y)$ |
| 270° CCW | $(x,y) \to (-y,x)$ |

> Note: If the rotation was clockwise, it would be different. But since not specified, assume counterclockwise.

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🧩 Final Answer:

To complete the task:

1. For each triangle, identify the coordinates of its vertices.
2. Apply the correct rotation rule:
- For 180°: negate both $x$ and $y$
- For 270°: apply $(-y, x)$
3. Plot the new points and draw the rotated triangle.

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💡 Tips for Drawing:

- Use graph paper carefully.
- Mark original points first.
- Apply transformation to each vertex.
- Connect the transformed points in the same order.

If you can provide exact coordinates or describe the shapes more precisely, I can give exact plotted answers.

Let me know if you'd like me to generate a visual explanation or help with plotting!