Question
Which sequence of transformations would result in a figure that is similar, but not congruent, to the original figure?
Select all that apply.
• a rotation about the origin of 25° followed by a reflection across the y-axis
• a reflection across the y-axis followed by a dilation with a scale factor of 0.5
• a dilation with a scale factor of 2 followed by a reflection across the x-axis
• a translation 2 units up followed by a rotation of 180° about the origin
Select all that apply.
• a rotation about the origin of 25° followed by a reflection across the y-axis
• a reflection across the y-axis followed by a dilation with a scale factor of 0.5
• a dilation with a scale factor of 2 followed by a reflection across the x-axis
• a translation 2 units up followed by a rotation of 180° about the origin
Answers
GPT-4o mini
To determine which sequences of transformations result in figures that are similar but not congruent to the original figure, we need to look for transformations that change the size of the figure while preserving the shape (but not making it identical in size).
1. **A rotation about the origin of 25° followed by a reflection across the y-axis:**
- Both rotation and reflection are rigid motions (isometries) that do not change the size of the figure. Therefore, this sequence will produce a figure that is congruent to the original, not similar but not congruent.
- **Not applicable.**
2. **A reflection across the y-axis followed by a dilation with a scale factor of 0.5:**
- The reflection across the y-axis is a rigid motion, but the dilation with a scale factor of 0.5 reduces the size of the figure. The resulting figure would be similar (same shape, different size).
- **Applicable.**
3. **A dilation with a scale factor of 2 followed by a reflection across the x-axis:**
- The dilation with a scale factor of 2 increases the size of the figure, and the subsequent reflection is a rigid motion. The resulting figure would also be similar.
- **Applicable.**
4. **A translation of 2 units up followed by a rotation of 180° about the origin:**
- Both the translation and the rotation are rigid motions that do not change the size or shape of the figure. Thus, the resulting figure will be congruent to the original.
- **Not applicable.**
In summary, the sequences of transformations that result in a figure that is similar but not congruent to the original figure are:
- A reflection across the y-axis followed by a dilation with a scale factor of 0.5
- A dilation with a scale factor of 2 followed by a reflection across the x-axis
1. **A rotation about the origin of 25° followed by a reflection across the y-axis:**
- Both rotation and reflection are rigid motions (isometries) that do not change the size of the figure. Therefore, this sequence will produce a figure that is congruent to the original, not similar but not congruent.
- **Not applicable.**
2. **A reflection across the y-axis followed by a dilation with a scale factor of 0.5:**
- The reflection across the y-axis is a rigid motion, but the dilation with a scale factor of 0.5 reduces the size of the figure. The resulting figure would be similar (same shape, different size).
- **Applicable.**
3. **A dilation with a scale factor of 2 followed by a reflection across the x-axis:**
- The dilation with a scale factor of 2 increases the size of the figure, and the subsequent reflection is a rigid motion. The resulting figure would also be similar.
- **Applicable.**
4. **A translation of 2 units up followed by a rotation of 180° about the origin:**
- Both the translation and the rotation are rigid motions that do not change the size or shape of the figure. Thus, the resulting figure will be congruent to the original.
- **Not applicable.**
In summary, the sequences of transformations that result in a figure that is similar but not congruent to the original figure are:
- A reflection across the y-axis followed by a dilation with a scale factor of 0.5
- A dilation with a scale factor of 2 followed by a reflection across the x-axis