Question
triangles ABC
and A'B'C'
are shown on the coordinate plane.
Which statement describes a sequence of transformations that can be used to verify that triangle
is similar to triangle
?
Reflect triangle ABC
across the x-axis, followed by a dilation with a scale factor of 2 about the origin.
Reflect triangle ABC
across the y-axis, followed by a dilation with a scale factor of 1/2
about the origin.
Reflect triangle ABC
across the x-axis, followed by a dilation with a scale factor of 1/2
about the origin.
Reflect triangle ABC
across the y-axis, followed by a dilation with a scale factor of 2 about the origin.
© 2025 |
and A'B'C'
are shown on the coordinate plane.
Which statement describes a sequence of transformations that can be used to verify that triangle
is similar to triangle
?
Reflect triangle ABC
across the x-axis, followed by a dilation with a scale factor of 2 about the origin.
Reflect triangle ABC
across the y-axis, followed by a dilation with a scale factor of 1/2
about the origin.
Reflect triangle ABC
across the x-axis, followed by a dilation with a scale factor of 1/2
about the origin.
Reflect triangle ABC
across the y-axis, followed by a dilation with a scale factor of 2 about the origin.
© 2025 |
Answers
Answered by
GPT-4o mini
To determine which sequence of transformations demonstrates that triangle ABC is similar to triangle A'B'C', we need to analyze the transformations and their effects on the triangle's dimensions and orientation.
### Transformations Overview:
1. **Reflection**: A reflection across an axis (x-axis or y-axis) flips the triangle over that axis but does not change its size or shape.
2. **Dilation**: A dilation changes the size of the triangle while maintaining its shape. A dilation with a scale factor greater than 1 enlarges the triangle, while a scale factor less than 1 reduces its size.
### Analyzing Options:
1. **Reflect triangle ABC across the x-axis, followed by a dilation with a scale factor of 2 about the origin**:
- This will flip triangle ABC over the x-axis (changing its orientation) and then enlarge it.
2. **Reflect triangle ABC across the y-axis, followed by a dilation with a scale factor of 1/2 about the origin**:
- This steps also flip the triangle (changing orientation) and reduces its size.
3. **Reflect triangle ABC across the x-axis, followed by a dilation with a scale factor of 1/2 about the origin**:
- Similar to option 1, but the size is reduced rather than enlarged.
4. **Reflect triangle ABC across the y-axis, followed by a dilation with a scale factor of 2 about the origin**:
- This reflects the triangle over the y-axis (changing its orientation) and enlarges it.
### Determining Similarity:
For triangles to be similar, they must have the same shape, which can be achieved by resizing (dilation) without bending or reshaping the figure. Reflection does not affect similarity in terms of shape but will affect the orientation.
To conclude, to verify similarity:
- If triangle A'B'C' is the same shape but larger or smaller than triangle ABC, using **a dilation with the correct scale factor** after a reflection (which preserves the shape) would be the correct transformation sequence.
Thus, **you should select the option that provides the correct scaling factor after reflecting**.
Without specific dimensions of triangles ABC and A'B'C', it's impossible to conclusively say which transformation leads to similarity. However, you should choose the option that maintains the proportional relationships between the sides (scale factor of 2 for enlargement or 1/2 for reduction) along with the shape-preserving reflection intended to align them.
If more specific information about triangles ABC and A'B'C' were provided, a definitive answer could be formulated.
### Transformations Overview:
1. **Reflection**: A reflection across an axis (x-axis or y-axis) flips the triangle over that axis but does not change its size or shape.
2. **Dilation**: A dilation changes the size of the triangle while maintaining its shape. A dilation with a scale factor greater than 1 enlarges the triangle, while a scale factor less than 1 reduces its size.
### Analyzing Options:
1. **Reflect triangle ABC across the x-axis, followed by a dilation with a scale factor of 2 about the origin**:
- This will flip triangle ABC over the x-axis (changing its orientation) and then enlarge it.
2. **Reflect triangle ABC across the y-axis, followed by a dilation with a scale factor of 1/2 about the origin**:
- This steps also flip the triangle (changing orientation) and reduces its size.
3. **Reflect triangle ABC across the x-axis, followed by a dilation with a scale factor of 1/2 about the origin**:
- Similar to option 1, but the size is reduced rather than enlarged.
4. **Reflect triangle ABC across the y-axis, followed by a dilation with a scale factor of 2 about the origin**:
- This reflects the triangle over the y-axis (changing its orientation) and enlarges it.
### Determining Similarity:
For triangles to be similar, they must have the same shape, which can be achieved by resizing (dilation) without bending or reshaping the figure. Reflection does not affect similarity in terms of shape but will affect the orientation.
To conclude, to verify similarity:
- If triangle A'B'C' is the same shape but larger or smaller than triangle ABC, using **a dilation with the correct scale factor** after a reflection (which preserves the shape) would be the correct transformation sequence.
Thus, **you should select the option that provides the correct scaling factor after reflecting**.
Without specific dimensions of triangles ABC and A'B'C', it's impossible to conclusively say which transformation leads to similarity. However, you should choose the option that maintains the proportional relationships between the sides (scale factor of 2 for enlargement or 1/2 for reduction) along with the shape-preserving reflection intended to align them.
If more specific information about triangles ABC and A'B'C' were provided, a definitive answer could be formulated.