Rotation of 3 Vertices around the Origin Starting in Quadrant I (A) - Free Printable

It looks like you've uploaded an image showing rotation answers for geometric shapes on a coordinate grid. The image contains four examples of rotating triangles about the origin $(0, 0)$ by $90^\circ$, either clockwise or counterclockwise.

Let’s go through each case and explain how to perform these rotations, as well as verify the correctness of the provided answers.

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🔷 Rotation Rules About the Origin (0, 0)

For any point $(x, y)$:

1. Rotate $90^\circ$ Clockwise:
$$(x, y) \rightarrow (y, -x)$$

2. Rotate $90^\circ$ Counterclockwise:
$$(x, y) \rightarrow (-y, x)$$

3. Rotate $180^\circ$:
$$(x, y) \rightarrow (-x, -y)$$

We'll use these rules to analyze each quadrant in your image.

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🟦 Top Left: Rotate $90^\circ$ Clockwise about (0,0)

- Original triangle has vertices roughly at:
- $A = (1, 3)$
- $B = (3, 3)$
- $C = (2, 1)$

Apply $90^\circ$ clockwise rule: $(x, y) \rightarrow (y, -x)$

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

Now check the red triangle (rotated image):
- Points appear to be at $(3, -1), (3, -3), (1, -2)$ → ✔ Matches!

✔ Correct

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🟨 Top Right: Rotate $90^\circ$ Clockwise about (0,0)

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

Apply $90^\circ$ clockwise: $(x, y) \rightarrow (y, -x)$

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

Red triangle points:
- Appears to be at $(2, -1), (2, -3), (4, -2)$ → ✔ Matches!

✔ Correct

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🟩 Bottom Left: Rotate $90^\circ$ Clockwise about (0,0)

Original triangle:
- $A = (1, 2)$
- $B = (2, 2)$
- $C = (1.5, 1)$

Apply $90^\circ$ clockwise: $(x, y) \rightarrow (y, -x)$

- $A = (1, 2) \rightarrow (2, -1)$
- $B = (2, 2) \rightarrow (2, -2)$
- $C = (1.5, 1) \rightarrow (1, -1.5)$

Red rotated triangle:
- Points seem to be at $(2, -1), (2, -2), (1, -1.5)$ → ✔ Matches!

✔ Correct

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🟪 Bottom Right: Rotate $90^\circ$ Counterclockwise about (0,0)

Original triangle:
- $A = (1, 2)$
- $B = (2, 2)$
- $C = (1.5, 1)$

Rule: $90^\circ$ counterclockwise → $(x, y) \rightarrow (-y, x)$

- $A = (1, 2) \rightarrow (-2, 1)$
- $B = (2, 2) \rightarrow (-2, 2)$
- $C = (1.5, 1) \rightarrow (-1, 1.5)$

Check red triangle:
- Points: $(-2, 1), (-2, 2), (-1, 1.5)$ → ✔ Matches!

✔ Correct

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✔ Final Answer Summary:

All four rotation transformations are correctly performed in the image. Each original shape is rotated accurately according to the specified direction and angle about the origin $(0, 0)$, using the correct transformation rules.

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📌 Key Takeaways:

| Rotation | Rule |
|--------|------|
| $90^\circ$ Clockwise | $(x, y) \rightarrow (y, -x)$ |
| $90^\circ$ Counterclockwise | $(x, y) \rightarrow (-y, x)$ |

These rules help rotate any point around the origin without needing to measure angles manually.

If you want me to generate similar problems or explain how to draw them step-by-step, let me know!