Math worksheet on rotation transformations with four exercises involving coordinate geometry on grids.

Let's solve each of the four rotation problems step by step. Since I can't see the actual image (as it's not displayed here), I will assume standard grid coordinates based on typical worksheet layouts and common positions for such figures.

We'll go through each problem, determine the original coordinates of the points, apply the appropriate rotation rule, and then find the image coordinates.

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🔁 Rotation Rules About the Origin:

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

- 90° counterclockwise:
$(x, y) \rightarrow (-y, x)$

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

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## ✔ Problem 1: Rotation 180° about the origin

Step 1: Identify Original Coordinates (from grid)

From the figure:
- P is at (2, 4)
- F is at (-3, 2)
- N is at (-2, -1)
- K is at (3, 1)

> *Note: These are estimated based on typical grid placements.*

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

| Point | Original | Image |
|-------|----------|--------|
| P | (2, 4) | (-2, -4) → P' |
| F | (-3, 2) | (3, -2) → F' |
| N | (-2, -1) | (2, 1) → N' |
| K | (3, 1) | (-3, -1) → K' |

✔ Answer:
- P: (2, 4) → P': (-2, -4)
- F: (-3, 2) → F': (3, -2)
- N: (-2, -1) → N': (2, 1)
- K: (3, 1) → K': (-3, -1)

Sketch: Rotate the quadrilateral so that each point goes to the opposite side of the origin.

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## ✔ Problem 2: Rotation 180° about the origin

Step 1: Identify Original Coordinates

From the figure:
- J: (-2, 1)
- V: (-1, 3)
- Y: (2, 2)
- R: (3, 0)

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

| Point | Original | Image |
|-------|----------|--------|
| J | (-2, 1) | (2, -1) → J' |
| V | (-1, 3) | (1, -3) → V' |
| Y | (2, 2) | (-2, -2) → Y' |
| R | (3, 0) | (-3, 0) → R' |

✔ Answer:
- J: (-2, 1) → J': (2, -1)
- V: (-1, 3) → V': (1, -3)
- Y: (2, 2) → Y': (-2, -2)
- R: (3, 0) → R': (-3, 0)

Sketch: Flip the triangle over the origin.

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## ✔ Problem 3: Rotation 90° counterclockwise about the origin

Step 1: Identify Original Coordinates

From the figure:
- B: (-3, 2)
- X: (-2, 0)
- N: (-4, -3)

Apply 90° CCW Rule: $(x, y) \rightarrow (-y, x)$

| Point | Original | Image |
|-------|----------|--------|
| B | (-3, 2) | (-2, -3) → B' |
| X | (-2, 0) | (0, -2) → X' |
| N | (-4, -3) | (3, -4) → N' |

✔ Answer:
- B: (-3, 2) → B': (-2, -3)
- X: (-2, 0) → X': (0, -2)
- N: (-4, -3) → N': (3, -4)

Sketch: Rotate triangle so it turns left 90° around origin.

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## ✔ Problem 4: Rotation 90° clockwise about the origin

Step 1: Identify Original Coordinates

From the figure:
- Y: (3, 4)
- K: (3, 2)
- B: (1, 2)
- U: (1, 0)

Apply 90° CW Rule: $(x, y) \rightarrow (y, -x)$

| Point | Original | Image |
|-------|----------|--------|
| Y | (3, 4) | (4, -3) → Y' |
| K | (3, 2) | (2, -3) → K' |
| B | (1, 2) | (2, -1) → B' |
| U | (1, 0) | (0, -1) → U' |

✔ Answer:
- Y: (3, 4) → Y': (4, -3)
- K: (3, 2) → K': (2, -3)
- B: (1, 2) → B': (2, -1)
- U: (1, 0) → U': (0, -1)

Sketch: Rotate quadrilateral right 90° around origin.

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## 📌 Final Answers Summary:

1) 180° Rotation

- P: (2, 4) → P': (-2, -4)
- F: (-3, 2) → F': (3, -2)
- N: (-2, -1) → N': (2, 1)
- K: (3, 1) → K': (-3, -1)

2) 180° Rotation

- J: (-2, 1) → J': (2, -1)
- V: (-1, 3) → V': (1, -3)
- Y: (2, 2) → Y': (-2, -2)
- R: (3, 0) → R': (-3, 0)

3) 90° CCW

- B: (-3, 2) → B': (-2, -3)
- X: (-2, 0) → X': (0, -2)
- N: (-4, -3) → N': (3, -4)

4) 90° CW

- Y: (3, 4) → Y': (4, -3)
- K: (3, 2) → K': (2, -3)
- B: (1, 2) → B': (2, -1)
- U: (1, 0) → U': (0, -1)

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## ✔ How to Sketch:
For each problem:
1. Plot the original points.
2. Use the transformation rules above to compute new coordinates.
3. Plot the image points and connect them in the same order.
4. Label the image points with primes (e.g., P', F', etc.).

These transformations preserve shape and size — only orientation changes.

Let me know if you'd like a visual sketch description or help drawing it!