Question
Figure G is rotated 90degree clockwise about the origin and then reflected over the x-axis, forming figure H.
On a coordinate plane, triangle G has points (negative 3, 1), (negative 1, 2), (negative 2, 5). Triangle H has points (2, negative 1), (1, negative 3), (5, negative 2).
Which sequence of transformations will produce the same results?
a reflection over the y-axis and then a rotation 90degree clockwise about the origin
a reflection over the x-axis and then a rotation 90degree clockwise about the origin
a 180degree rotation about the origin
a reflection over the y-axis and then a reflection over the x-axis
On a coordinate plane, triangle G has points (negative 3, 1), (negative 1, 2), (negative 2, 5). Triangle H has points (2, negative 1), (1, negative 3), (5, negative 2).
Which sequence of transformations will produce the same results?
a reflection over the y-axis and then a rotation 90degree clockwise about the origin
a reflection over the x-axis and then a rotation 90degree clockwise about the origin
a 180degree rotation about the origin
a reflection over the y-axis and then a reflection over the x-axis
Answers
GPT-4o mini
To determine which sequence of transformations produces the same result as rotating triangle G 90 degrees clockwise about the origin and then reflecting it over the x-axis, we can break down the transformations step by step.
1. **Original Points of Triangle G:**
- \( G(-3, 1) \)
- \( G(-1, 2) \)
- \( G(-2, 5) \)
2. **Transformation 1: Rotate 90 degrees clockwise about the origin**
- The formula for a 90-degree clockwise rotation of a point \( (x, y) \) is \( (y, -x) \).
- Applying this transformation:
- For \( (-3, 1) \): \( (1, 3) \)
- For \( (-1, 2) \): \( (2, 1) \)
- For \( (-2, 5) \): \( (5, 2) \)
So after rotation, the new coordinates are:
- \( (1, 3) \)
- \( (2, 1) \)
- \( (5, 2) \)
3. **Transformation 2: Reflect over the x-axis**
- The formula for reflecting a point \( (x, y) \) over the x-axis is \( (x, -y) \).
- Applying this transformation:
- For \( (1, 3) \): \( (1, -3) \)
- For \( (2, 1) \): \( (2, -1) \)
- For \( (5, 2) \): \( (5, -2) \)
So after reflection, the final coordinates will be:
- \( (1, -3) \)
- \( (2, -1) \)
- \( (5, -2) \)
Now, let's see which option gives the same result as these transformations:
**Examining the Options:**
1. **A reflection over the y-axis, then a rotation 90 degrees clockwise about the origin:**
- Reflecting \( G(-3, 1) \) over the y-axis results in \( (3, 1) \).
- Rotating \( (3, 1) \) 90 degrees clockwise results in \( (1, -3) \).
- This transformation matches the first point of H.
_We would continue this process for the other points, but if this transformation already matches one point, let's check the others to gather more insights._
2. **A reflection over the x-axis, then a rotation 90 degrees clockwise about the origin:**
- Reflecting \( (-3, 1) \) over the x-axis results in \( (-3, -1) \).
- Rotating \( (-3, -1) \) 90 degrees clockwise results in \( (-1, 3) \).
- This point does not match H.
3. **A 180-degree rotation about the origin:**
- Rotating \( (-3, 1) \) 180 degrees results in \( (3, -1) \).
- This point does not match H.
4. **A reflection over the y-axis, then a reflection over the x-axis:**
- Reflecting \( (-3, 1) \) over the y-axis gives \( (3, 1) \).
- Reflecting \( (3, 1) \) over the x-axis gives \( (3, -1) \).
- This transformation does not match H.
From analysis, the transformation that gives points that align with triangle H is **option 1: a reflection over the y-axis and then a rotation 90 degrees clockwise about the origin.**
1. **Original Points of Triangle G:**
- \( G(-3, 1) \)
- \( G(-1, 2) \)
- \( G(-2, 5) \)
2. **Transformation 1: Rotate 90 degrees clockwise about the origin**
- The formula for a 90-degree clockwise rotation of a point \( (x, y) \) is \( (y, -x) \).
- Applying this transformation:
- For \( (-3, 1) \): \( (1, 3) \)
- For \( (-1, 2) \): \( (2, 1) \)
- For \( (-2, 5) \): \( (5, 2) \)
So after rotation, the new coordinates are:
- \( (1, 3) \)
- \( (2, 1) \)
- \( (5, 2) \)
3. **Transformation 2: Reflect over the x-axis**
- The formula for reflecting a point \( (x, y) \) over the x-axis is \( (x, -y) \).
- Applying this transformation:
- For \( (1, 3) \): \( (1, -3) \)
- For \( (2, 1) \): \( (2, -1) \)
- For \( (5, 2) \): \( (5, -2) \)
So after reflection, the final coordinates will be:
- \( (1, -3) \)
- \( (2, -1) \)
- \( (5, -2) \)
Now, let's see which option gives the same result as these transformations:
**Examining the Options:**
1. **A reflection over the y-axis, then a rotation 90 degrees clockwise about the origin:**
- Reflecting \( G(-3, 1) \) over the y-axis results in \( (3, 1) \).
- Rotating \( (3, 1) \) 90 degrees clockwise results in \( (1, -3) \).
- This transformation matches the first point of H.
_We would continue this process for the other points, but if this transformation already matches one point, let's check the others to gather more insights._
2. **A reflection over the x-axis, then a rotation 90 degrees clockwise about the origin:**
- Reflecting \( (-3, 1) \) over the x-axis results in \( (-3, -1) \).
- Rotating \( (-3, -1) \) 90 degrees clockwise results in \( (-1, 3) \).
- This point does not match H.
3. **A 180-degree rotation about the origin:**
- Rotating \( (-3, 1) \) 180 degrees results in \( (3, -1) \).
- This point does not match H.
4. **A reflection over the y-axis, then a reflection over the x-axis:**
- Reflecting \( (-3, 1) \) over the y-axis gives \( (3, 1) \).
- Reflecting \( (3, 1) \) over the x-axis gives \( (3, -1) \).
- This transformation does not match H.
From analysis, the transformation that gives points that align with triangle H is **option 1: a reflection over the y-axis and then a rotation 90 degrees clockwise about the origin.**