Question
Quadrilateral QRST and its image, Q’R’S’T’, are shown. What is the sequence of transformations that results in quadrilateral QRST being mapped onto quadrilateral Q’R’S’T’?
(10 Points)
Reflection over the x-axis, followed by a dilation about the origin by a scale factor of 2
90° clockwise rotation about the origin, followed by a dilation about the origin by a scale factor of 2
Dilation about the origin by a scale factor of 2, followed by a translation 2 units right and then 2 units down
270° counterclockwise rotation about the origin, followed by a dilation about the origin by a scale factor of 2
(10 Points)
Reflection over the x-axis, followed by a dilation about the origin by a scale factor of 2
90° clockwise rotation about the origin, followed by a dilation about the origin by a scale factor of 2
Dilation about the origin by a scale factor of 2, followed by a translation 2 units right and then 2 units down
270° counterclockwise rotation about the origin, followed by a dilation about the origin by a scale factor of 2
Answers
GPT-4o mini
To determine the sequence of transformations that maps quadrilateral QRST onto its image Q’R’S’T’, we can analyze the options given in the context of typical transformations: reflections, rotations, dilations, and translations.
1. **Reflection over the x-axis** - This transformation flips the figure over the x-axis. We would need to check if this transformation alone could cause the corresponding vertices to match.
2. **90° clockwise rotation about the origin** - This would turn the quadrilateral, but we would need to analyze how the coordinates change to see if they match Q’R’S’T’.
3. **Dilation about the origin by a scale factor of 2** - This transformation would enlarge the figure but not rotate or translate it.
4. **Translation 2 units right and then 2 units down** - This would shift the quadrilateral in a linear path, but again, we would need to see how these movements impact the corresponding coordinates.
5. **270° counterclockwise rotation about the origin** - This is equivalent to a 90° clockwise rotation and would rotate the figure but may or may not match the new coordinates of Q’R’S’T’.
6. **Dilation followed by a translation** - This combines the enlargement followed by movement, potentially leading to a match depending on the scale factor and translation distance.
Analyzing these transformations typically requires knowledge of the specific coordinates of the points in QRST and Q’R’S’T’. However, without the actual image or the coordinates, I would suggest the most common transformation style for enlarging and translating an image is:
- **Dilation about the origin by a scale factor of 2, followed by a translation** (likely to position the dilated shape correctly).
If the vertices of Q’R’S’T’ are positioned correctly with respect to QRST after dilation, then it would reflect the actions resulting in a match.
So, depending on the intentions here, the most plausible sequence is:
**Dilation about the origin by a scale factor of 2, followed by a translation 2 units right and then 2 units down.**
This sequence allows for size change and position adjustment to align the two quadrilaterals.
1. **Reflection over the x-axis** - This transformation flips the figure over the x-axis. We would need to check if this transformation alone could cause the corresponding vertices to match.
2. **90° clockwise rotation about the origin** - This would turn the quadrilateral, but we would need to analyze how the coordinates change to see if they match Q’R’S’T’.
3. **Dilation about the origin by a scale factor of 2** - This transformation would enlarge the figure but not rotate or translate it.
4. **Translation 2 units right and then 2 units down** - This would shift the quadrilateral in a linear path, but again, we would need to see how these movements impact the corresponding coordinates.
5. **270° counterclockwise rotation about the origin** - This is equivalent to a 90° clockwise rotation and would rotate the figure but may or may not match the new coordinates of Q’R’S’T’.
6. **Dilation followed by a translation** - This combines the enlargement followed by movement, potentially leading to a match depending on the scale factor and translation distance.
Analyzing these transformations typically requires knowledge of the specific coordinates of the points in QRST and Q’R’S’T’. However, without the actual image or the coordinates, I would suggest the most common transformation style for enlarging and translating an image is:
- **Dilation about the origin by a scale factor of 2, followed by a translation** (likely to position the dilated shape correctly).
If the vertices of Q’R’S’T’ are positioned correctly with respect to QRST after dilation, then it would reflect the actions resulting in a match.
So, depending on the intentions here, the most plausible sequence is:
**Dilation about the origin by a scale factor of 2, followed by a translation 2 units right and then 2 units down.**
This sequence allows for size change and position adjustment to align the two quadrilaterals.