Lesson 6 Practice Problems 1. Find a sequence of rigid motions and square $$ E F G H $$.

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It looks like you're working on a geometry problem involving rigid motions and a square labeled $$ EFGH $$. However, the problem statement appears to be incomplete or might be referencing a figure that isn't provided here. To effectively help you, I'll provide a general explanation of how to approach finding a sequence of rigid motions to transform a figure into square $$ EFGH $$.

**Rigid Motions Overview:**
Rigid motions in geometry include transformations that preserve the shape and size of figures. The main types of rigid motions are:

1. **Translation:** Sliding the figure without rotating or flipping.
2. **Rotation:** Turning the figure around a fixed point.
3. **Reflection:** Flipping the figure over a line (mirror image).
4. **Glide Reflection:** A combination of a reflection and a translation along the direction of the reflection line.

**General Steps to Find a Sequence of Rigid Motions:**

1. **Identify the Initial and Final Positions:**
   - Determine the position of the original figure and the position of square $$ EFGH $$.

2. **Determine the Necessary Transformations:**
   - **Translation:** If $$ EFGH $$ has been moved from its original position without rotation or flipping, use a translation vector that moves each point of the original figure to its corresponding point in $$ EFGH $$.
   - **Rotation:** If $$ EFGH $$ is rotated relative to the original figure, identify the center of rotation and the angle of rotation.
   - **Reflection:** If $$ EFGH $$ is a mirror image of the original figure, determine the line of reflection.
   - **Glide Reflection:** If both a reflection and a translation are needed, combine these two transformations.

3. **Apply the Transformations in Sequence:**
   - The order of transformations matters. Typically, reflections and rotations are applied before translations.
   - For example, first rotate the figure, then reflect it, and finally translate it to the desired position.

**Example Problem:**

*Suppose you have a square $$ ABCD $$ that you want to transform into square $$ EFGH $$. Let's say square $$ EFGH $$ is the image of $$ ABCD $$ after a 90-degree rotation about the origin, followed by a translation 5 units to the right.*

**Solution:**

1. **Rotation:**
   - Rotate square $$ ABCD $$ 90 degrees clockwise about the origin to get an intermediate square $$ A'B'C'D' $$.

2. **Translation:**
   - Translate square $$ A'B'C'D' $$ 5 units to the right along the x-axis to arrive at square $$ EFGH $$.

**Sequence of Rigid Motions:**
1. **Rotate 90° clockwise about the origin.**
2. **Translate 5 units to the right.**

**Visual Representation:**

1. Original Square $$ ABCD $$
2. After Rotation → Square $$ A'B'C'D' $$
3. After Translation → Square $$ EFGH $$

**Tips:**

- **Identify Fixed Points:** Determine if there's a fixed point like the center of rotation.
- **Check Orientation:** Ensure that the orientation (clockwise or counterclockwise) matches between the original and transformed figures.
- **Use Coordinate Geometry:** Assign coordinates to key points to make calculations easier.

If you have a specific figure or more details about the positions and transformations involved with square $$ EFGH $$, feel free to share them, and I can provide a more tailored solution!
To transform the original square into square $$ EFGH $$, perform a 90-degree clockwise rotation about the origin followed by a 5-unit translation to the right.

Lesson 6 Practice Problems 1. Find a sequence of rigid motions and square \( E F G H \).

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