Question
Alexi transformed figure L such that its image is figure K after a 90degree counterclockwise rotation about the origin and a reflection over the y-axis.
On a coordinate plane, triangle K has points (negative 3, 1), (negative 1, 2), (negative 2, 5). Triangle L has points (2, negative 1), (1, negative 3), (5, negative 2).
Jonathan used figure L to perform the same transformation, but he reflected figure L over the y-axis before performing the 90degree counterclockwise rotation. Which best compares Alexi’s and Jonathan’s transformed figures?
The change in order will not affect the final image. Their transformed figures will be identical.
The change in order will result in the transformed images having vertices in the same location, but in a different orientation.
The change in order will result in the transformed image of figure L being transformed back onto figure L.
The change in order will result in the transformed image of figure L having a different location.
On a coordinate plane, triangle K has points (negative 3, 1), (negative 1, 2), (negative 2, 5). Triangle L has points (2, negative 1), (1, negative 3), (5, negative 2).
Jonathan used figure L to perform the same transformation, but he reflected figure L over the y-axis before performing the 90degree counterclockwise rotation. Which best compares Alexi’s and Jonathan’s transformed figures?
The change in order will not affect the final image. Their transformed figures will be identical.
The change in order will result in the transformed images having vertices in the same location, but in a different orientation.
The change in order will result in the transformed image of figure L being transformed back onto figure L.
The change in order will result in the transformed image of figure L having a different location.
Answers
GPT-4o mini
To compare Alexi's and Jonathan's transformations, we need to analyze the effects of their respective transformations step-by-step.
**Alexi's Transformation:**
1. **90-degree counterclockwise rotation about the origin.**
- The rotation of a point \((x, y)\) by 90 degrees counterclockwise results in the new coordinates \((-y, x)\).
2. **Reflection over the y-axis.**
- Reflecting a point \((x, y)\) over the y-axis changes the coordinates to \((-x, y)\).
Let's apply Alexi's transformation to triangle L with vertices at \((2, -1)\), \((1, -3)\), and \((5, -2)\).
- For the point \((2, -1)\):
1. Rotate: \((-(-1), 2) = (1, 2)\)
2. Reflect: \((-1, 2)\)
- For the point \((1, -3)\):
1. Rotate: \((-(-3), 1) = (3, 1)\)
2. Reflect: \((-3, 1)\)
- For the point \((5, -2)\):
1. Rotate: \((-(-2), 5) = (2, 5)\)
2. Reflect: \((-2, 5)\)
Thus, the transformed coordinates of triangle L after Alexi's transformation are:
- \((-1, 2)\)
- \((-3, 1)\)
- \((-2, 5)\)
**Jonathan's Transformation:**
1. **Reflection over the y-axis.**
- This step remains the same: the coordinates become \((-x, y)\) from \((x, y)\).
2. **90-degree counterclockwise rotation about the origin.**
- Now let's apply this to the reflected triangle.
For triangle L, reflecting over the y-axis gives:
- \((-2, -1)\)
- \((-1, -3)\)
- \((-5, -2)\)
Now let's apply the 90-degree counterclockwise rotation:
- For the point \((-2, -1)\):
1. Rotate: \((-(-1), -2) = (1, -2)\)
- For the point \((-1, -3)\):
1. Rotate: \((-(-3), -1) = (3, -1)\)
- For the point \((-5, -2)\):
1. Rotate: \((-(-2), -5) = (2, -5)\)
This results in transformed coordinates of triangle L after Jonathan's transformation:
- \((1, -2)\)
- \((3, -1)\)
- \((2, -5)\)
**Comparison:**
Now, we compare the transformed figures from each individual:
- Alexi’s transformation resulted in points \((-1, 2)\), \((-3, 1)\), and \((-2, 5)\).
- Jonathan’s transformation resulted in points \((1, -2)\), \((3, -1)\), and \((2, -5)\).
Seeing that these sets of points are different, we can conclude:
The change in order will result in the transformed images having vertices in the same location, but in a different orientation.
Therefore, the correct answer is:
**The change in order will result in the transformed images having vertices in the same location, but in a different orientation.**
**Alexi's Transformation:**
1. **90-degree counterclockwise rotation about the origin.**
- The rotation of a point \((x, y)\) by 90 degrees counterclockwise results in the new coordinates \((-y, x)\).
2. **Reflection over the y-axis.**
- Reflecting a point \((x, y)\) over the y-axis changes the coordinates to \((-x, y)\).
Let's apply Alexi's transformation to triangle L with vertices at \((2, -1)\), \((1, -3)\), and \((5, -2)\).
- For the point \((2, -1)\):
1. Rotate: \((-(-1), 2) = (1, 2)\)
2. Reflect: \((-1, 2)\)
- For the point \((1, -3)\):
1. Rotate: \((-(-3), 1) = (3, 1)\)
2. Reflect: \((-3, 1)\)
- For the point \((5, -2)\):
1. Rotate: \((-(-2), 5) = (2, 5)\)
2. Reflect: \((-2, 5)\)
Thus, the transformed coordinates of triangle L after Alexi's transformation are:
- \((-1, 2)\)
- \((-3, 1)\)
- \((-2, 5)\)
**Jonathan's Transformation:**
1. **Reflection over the y-axis.**
- This step remains the same: the coordinates become \((-x, y)\) from \((x, y)\).
2. **90-degree counterclockwise rotation about the origin.**
- Now let's apply this to the reflected triangle.
For triangle L, reflecting over the y-axis gives:
- \((-2, -1)\)
- \((-1, -3)\)
- \((-5, -2)\)
Now let's apply the 90-degree counterclockwise rotation:
- For the point \((-2, -1)\):
1. Rotate: \((-(-1), -2) = (1, -2)\)
- For the point \((-1, -3)\):
1. Rotate: \((-(-3), -1) = (3, -1)\)
- For the point \((-5, -2)\):
1. Rotate: \((-(-2), -5) = (2, -5)\)
This results in transformed coordinates of triangle L after Jonathan's transformation:
- \((1, -2)\)
- \((3, -1)\)
- \((2, -5)\)
**Comparison:**
Now, we compare the transformed figures from each individual:
- Alexi’s transformation resulted in points \((-1, 2)\), \((-3, 1)\), and \((-2, 5)\).
- Jonathan’s transformation resulted in points \((1, -2)\), \((3, -1)\), and \((2, -5)\).
Seeing that these sets of points are different, we can conclude:
The change in order will result in the transformed images having vertices in the same location, but in a different orientation.
Therefore, the correct answer is:
**The change in order will result in the transformed images having vertices in the same location, but in a different orientation.**