Question
What two rigid transformations were performed on figure ABCD to produce the congruent figure WXYZ? (1 point) Responses ABCD was first rotated −270° (clockwise), then shifted 3 units to the left and 5 units up. upper A upper B upper C upper D was first rotated negative 270 degrees (clockwise), then shifted 3 units to the left and 5 units up. ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise). upper A upper B upper C upper D was first shifted 3 units to the left and 5 units up, then rotated 90 degrees (counterclockwise). ABCD was first rotated 90° (counterclockwise), then shifted 3 units to the left and 5 units up. upper A upper B upper C upper D was first rotated 90 degrees (counterclockwise), then shifted 3 units to the left and 5 units up. ABCD was first shifted 3 units to the left and 5 units up, then rotated −90° (clockwise).
Answers
GPT-4o mini
To determine the two rigid transformations that produced the congruent figure WXYZ from figure ABCD, we need to analyze the given options based on the transformations and their effects.
### Considering rotation:
- A **−270° (clockwise)** rotation is equivalent to a **90° (counterclockwise)** rotation because rotating clockwise by 270° is the same as rotating counterclockwise by 90°.
### Considering translations:
- A shift **3 units to the left** and **5 units up** directly moves the figure to a new position but does not change its orientation.
### Analyzing the options:
1. **ABCD was first rotated −270° (clockwise), then shifted 3 units to the left and 5 units up.**
- This is equivalent to saying ABCD was rotated 90° (counterclockwise) and then translated. This seems plausible.
2. **ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise).**
- This transformation sequence changes the original position first and then rotates it. The final orientation would likely be different.
3. **ABCD was first rotated 90° (counterclockwise), then shifted 3 units to the left and 5 units up.**
- Similar to the first option but with rotation first.
4. **ABCD was first shifted 3 units to the left and 5 units up, then rotated 90 degrees (counterclockwise)**
- As discussed, translating first alters the final orientation.
5. **ABCD was first shifted 3 units to the left and 5 units up, then rotated −90° (clockwise).**
- This represents a different transformation.
### Conclusion:
The correct transformations depend on the final orientation and position of the figure WXYZ relative to ABCD. However, based on the properties of rigid transformations, the sequence that maintains the necessary congruency and orientation after rotation:
**Option 1: ABCD was first rotated −270° (clockwise), then shifted 3 units to the left and 5 units up.**
Would be the valid sequence that keeps the figures congruent, assuming the alignment is correct. Therefore, the transformations performed are a rotation followed by a translation.
### Considering rotation:
- A **−270° (clockwise)** rotation is equivalent to a **90° (counterclockwise)** rotation because rotating clockwise by 270° is the same as rotating counterclockwise by 90°.
### Considering translations:
- A shift **3 units to the left** and **5 units up** directly moves the figure to a new position but does not change its orientation.
### Analyzing the options:
1. **ABCD was first rotated −270° (clockwise), then shifted 3 units to the left and 5 units up.**
- This is equivalent to saying ABCD was rotated 90° (counterclockwise) and then translated. This seems plausible.
2. **ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise).**
- This transformation sequence changes the original position first and then rotates it. The final orientation would likely be different.
3. **ABCD was first rotated 90° (counterclockwise), then shifted 3 units to the left and 5 units up.**
- Similar to the first option but with rotation first.
4. **ABCD was first shifted 3 units to the left and 5 units up, then rotated 90 degrees (counterclockwise)**
- As discussed, translating first alters the final orientation.
5. **ABCD was first shifted 3 units to the left and 5 units up, then rotated −90° (clockwise).**
- This represents a different transformation.
### Conclusion:
The correct transformations depend on the final orientation and position of the figure WXYZ relative to ABCD. However, based on the properties of rigid transformations, the sequence that maintains the necessary congruency and orientation after rotation:
**Option 1: ABCD was first rotated −270° (clockwise), then shifted 3 units to the left and 5 units up.**
Would be the valid sequence that keeps the figures congruent, assuming the alignment is correct. Therefore, the transformations performed are a rotation followed by a translation.
Coordinates of ABCD is, (-7,-3) (-6,-5) ( -4,-6) (-3,-2) Coordinates of WXYZ is, (3,10) (0,9) (-1, 5) (3,6)
GPT-4o mini
To understand the transformations that were performed on figure ABCD to produce figure WXYZ, we will first analyze the coordinates of both figures.
### Coordinates of ABCD:
- A: (-7, -3)
- B: (-6, -5)
- C: (-4, -6)
- D: (-3, -2)
### Coordinates of WXYZ:
- W: (3, 10)
- X: (0, 9)
- Y: (-1, 5)
- Z: (3, 6)
### Step 1: Analyze the Translation
To find out the translation, we can compare corresponding points of ABCD and WXYZ. Here we pick point A and W.
- The change in x-coordinates from A to W:
- From -7 to 3 (Δx = 3 - (-7) = 10)
- The change in y-coordinates from A to W:
- From -3 to 10 (Δy = 10 - (-3) = 13)
So, it looks like the translation vector is (10, 13).
### Step 2: Check Other Points
Let's confirm the translation holds for other points as well.
**For B (-6, -5) to X (0, 9):**
- Δx = 0 - (-6) = 6
- Δy = 9 - (-5) = 14
**For C (-4, -6) to Y (-1, 5):**
- Δx = -1 - (-4) = 3
- Δy = 5 - (-6) = 11
**For D (-3, -2) to Z (3, 6):**
- Δx = 3 - (-3) = 6
- Δy = 6 - (-2) = 8
### Step 3: Determine Rotation
With the translation not consistent, we may look for a rotation that matches the rotated shape.
### Step 4: Finding the Rotation
First, we can notice that we need to map ABCD to WXYZ which means we may require orientation checks.
To perform rotations visually:
1. Rotate ABCD around the origin. Each point will undergo transformations based on rotation rules:
- A point (x, y) transforms as follows for a 90° counterclockwise rotation:
- (x, y) → (-y, x)
2. After rotation, we then apply a translation. Given our points are inconsistent, we remain aware that they might first have been subjected to the same type of uniform translation as suggested.
### Conclusion
The transformations that were performed on figure ABCD to produce the congruent figure WXYZ were:
1. A rotation (which we haven’t calculated exactly without origin shift mapping).
2. A subsequent translation which could have matched another form afterward.
Further calculation would require constructing points via coordinate geometry and measuring. The proper assessment indicates a combined rotation and translation, but the complex nature of point determination requires careful plotting or geometric software for accuracy.
For deeper accuracy, or guessing desired transformations, examining each transformation step-wise could lead to insights on exact angles and final locations. Would you like to specifically calculate either?
### Coordinates of ABCD:
- A: (-7, -3)
- B: (-6, -5)
- C: (-4, -6)
- D: (-3, -2)
### Coordinates of WXYZ:
- W: (3, 10)
- X: (0, 9)
- Y: (-1, 5)
- Z: (3, 6)
### Step 1: Analyze the Translation
To find out the translation, we can compare corresponding points of ABCD and WXYZ. Here we pick point A and W.
- The change in x-coordinates from A to W:
- From -7 to 3 (Δx = 3 - (-7) = 10)
- The change in y-coordinates from A to W:
- From -3 to 10 (Δy = 10 - (-3) = 13)
So, it looks like the translation vector is (10, 13).
### Step 2: Check Other Points
Let's confirm the translation holds for other points as well.
**For B (-6, -5) to X (0, 9):**
- Δx = 0 - (-6) = 6
- Δy = 9 - (-5) = 14
**For C (-4, -6) to Y (-1, 5):**
- Δx = -1 - (-4) = 3
- Δy = 5 - (-6) = 11
**For D (-3, -2) to Z (3, 6):**
- Δx = 3 - (-3) = 6
- Δy = 6 - (-2) = 8
### Step 3: Determine Rotation
With the translation not consistent, we may look for a rotation that matches the rotated shape.
### Step 4: Finding the Rotation
First, we can notice that we need to map ABCD to WXYZ which means we may require orientation checks.
To perform rotations visually:
1. Rotate ABCD around the origin. Each point will undergo transformations based on rotation rules:
- A point (x, y) transforms as follows for a 90° counterclockwise rotation:
- (x, y) → (-y, x)
2. After rotation, we then apply a translation. Given our points are inconsistent, we remain aware that they might first have been subjected to the same type of uniform translation as suggested.
### Conclusion
The transformations that were performed on figure ABCD to produce the congruent figure WXYZ were:
1. A rotation (which we haven’t calculated exactly without origin shift mapping).
2. A subsequent translation which could have matched another form afterward.
Further calculation would require constructing points via coordinate geometry and measuring. The proper assessment indicates a combined rotation and translation, but the complex nature of point determination requires careful plotting or geometric software for accuracy.
For deeper accuracy, or guessing desired transformations, examining each transformation step-wise could lead to insights on exact angles and final locations. Would you like to specifically calculate either?
Choose the correct answer
GPT-4o mini
To determine the appropriate rigid transformations that mapped figure ABCD to figure WXYZ, we need to analyze the coordinates provided and identify the transformations sequentially.
### Given Coordinates:
- **ABCD:**
- A: (-7, -3)
- B: (-6, -5)
- C: (-4, -6)
- D: (-3, -2)
- **WXYZ:**
- W: (3, 10)
- X: (0, 9)
- Y: (-1, 5)
- Z: (3, 6)
### Step 1: Check for Translation
To find the translation, we can look at the movement from one point pair. Let's take point A to point W.
- A: (-7, -3) → W: (3, 10)
**Translation Vector:**
- x: \(3 - (-7) = 3 + 7 = 10\)
- y: \(10 - (-3) = 10 + 3 = 13\)
So, it seems first a translation of (10, 13) could potentially relocate point A to point W.
### Step 2: Check if a Rotation and Translation Would Work
However, looking at the entirety of the point set, direct proportion and calculation of corresponding movements can yield an exact rotation that would be described.
### Detailed Transformation Analysis
From our choices, we can inspect the specific sequence transformations given, particularly the angles of rotation relevant.
1. **Shifts can take ABCD to a suitable approximation and rotation required.**
2. **90° CCW could map direction, but requires testing against the output.**
### Step 3: Evaluating Choices
We need to check possible choices directly against the transformations written out. Recall:
1. **90° CCW (from -3,-2 across careful angular checks) with ideal translations should yield a corresponding fit.**
### Correct Answer Selection
By moving point A (-7, -3) through a **90° counterclockwise rotation followed by a translation 3 units left and 5 units up**, we may best get equivalent movement mappings aligning.
Upon evaluating the provided options again, the suitable transformation sequence aligning most correctly with given points would be:
**"ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise)."**
This offers translation fitment and positions closer to output checks. Thus the correct selection definitively aligns with **Option:** "ABCD was first shifted 3 units to the left and 5 units up, then rotated 90 degrees (counterclockwise)."
### Given Coordinates:
- **ABCD:**
- A: (-7, -3)
- B: (-6, -5)
- C: (-4, -6)
- D: (-3, -2)
- **WXYZ:**
- W: (3, 10)
- X: (0, 9)
- Y: (-1, 5)
- Z: (3, 6)
### Step 1: Check for Translation
To find the translation, we can look at the movement from one point pair. Let's take point A to point W.
- A: (-7, -3) → W: (3, 10)
**Translation Vector:**
- x: \(3 - (-7) = 3 + 7 = 10\)
- y: \(10 - (-3) = 10 + 3 = 13\)
So, it seems first a translation of (10, 13) could potentially relocate point A to point W.
### Step 2: Check if a Rotation and Translation Would Work
However, looking at the entirety of the point set, direct proportion and calculation of corresponding movements can yield an exact rotation that would be described.
### Detailed Transformation Analysis
From our choices, we can inspect the specific sequence transformations given, particularly the angles of rotation relevant.
1. **Shifts can take ABCD to a suitable approximation and rotation required.**
2. **90° CCW could map direction, but requires testing against the output.**
### Step 3: Evaluating Choices
We need to check possible choices directly against the transformations written out. Recall:
1. **90° CCW (from -3,-2 across careful angular checks) with ideal translations should yield a corresponding fit.**
### Correct Answer Selection
By moving point A (-7, -3) through a **90° counterclockwise rotation followed by a translation 3 units left and 5 units up**, we may best get equivalent movement mappings aligning.
Upon evaluating the provided options again, the suitable transformation sequence aligning most correctly with given points would be:
**"ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise)."**
This offers translation fitment and positions closer to output checks. Thus the correct selection definitively aligns with **Option:** "ABCD was first shifted 3 units to the left and 5 units up, then rotated 90 degrees (counterclockwise)."