Geometry worksheet on rigid transformations featuring four coordinate plane diagrams illustrating triangle transformations.

Since I can't view or access images directly, I can’t see the specific figures or transformations in your uploaded image. However, based on the description of the worksheet you provided — "Rigid Transformations" with multiple coordinate plane diagrams showing pre-images and images of triangles — I can guide you through how to solve this type of problem and explain the reasoning.

---

🔷 What Are Rigid Transformations?

Rigid transformations are transformations that preserve the size and shape of a figure. The three types are:

1. Translation – sliding a figure without rotating or flipping.
2. Rotation – turning a figure around a fixed point.
3. Reflection – flipping a figure over a line.

A combination of these may be used.

---

🔷 How to Solve These Problems

For each question (e.g., "Describe the combination of rigid transformations that maps △ABC to △A'B'C'"), follow these steps:

#### Step 1: Identify the Coordinates
- Write down the coordinates of the vertices of the original triangle (pre-image) and the transformed triangle (image).
- Example:
- △ABC: A(1,1), B(3,1), C(1,4)
- △A'B'C': A'(4,4), B'(6,4), C'(4,7)

#### Step 2: Look for Patterns
Compare the coordinates to determine the transformation(s):

- If all points move the same distance in the same direction, it’s a translation.
- Example: Each x increases by 3, y increases by 3 → translation by ⟨3,3⟩
- If the figure is flipped over a line (like x-axis, y-axis, or y=x), it’s a reflection.
- If the figure rotates around a point, check angles and distances from a center (often origin or vertex).

#### Step 3: Check for Multiple Steps
Sometimes one transformation isn’t enough. You might need:
- A reflection followed by a translation
- Or a rotation followed by a reflection

Look at orientation:
- If the order of the vertices changes (e.g., clockwise vs counterclockwise), rotation or reflection likely occurred.

---

🔷 Let's Work Through an Example

Suppose in Problem #1:
- △ABC has points A(1,1), B(3,1), C(1,4)
- △A'B'C' has points A'(4,4), B'(6,4), C'(4,7)

→ All x-coordinates increased by 3, y-coordinates increased by 3
→ So, translation by ⟨3,3⟩

Answer: △ABC is translated by ⟨3,3⟩ to form △A'B'C'.

---

🔷 Common Transformation Descriptions

| Type | Description |
|------|-------------|
| Translation | “Translated by ⟨a,b⟩” or “Moved a units right and b units up” |
| Reflection | “Reflected over the x-axis”, “y-axis”, or “line y = x” |
| Rotation | “Rotated 90° counterclockwise about the origin” |
| Combination | “First reflected over the y-axis, then translated 2 units up” |

---

🔷 Tips for the Worksheet

For each of the four problems:

1. Label the points on both triangles.
2. Compare coordinates.
3. Determine if the figure flipped, turned, or slid.
4. Write a clear sentence describing the transformation(s).

---

🔷 Example Answer Format

> "△ABC is reflected over the y-axis and then translated 2 units up to obtain △A'B'C'."

Or:

> "△DEF is rotated 90° clockwise about the origin to get △D'E'F'."

---

📌 Final Advice

If you can describe what you see in the image (e.g., “Triangle ABC is in the bottom-left, and A’B’C’ is in the top-right”), I can help you solve it step-by-step.

Alternatively, you can provide the coordinates of the points for each triangle, and I’ll help you determine the transformation.

Let me know!