Question
Triangle ABC has the points A(1,1), B(7,2) and C(5,7). What composition transformation will result in △A"B"C"
being at A" (2 ,-1), B" (-4,0) and C" (-2,5)?(1 point)
Responses
a rotation of - 270 degrees and a move right by 3 and up by 2
a rotation of - 270 degrees and a move right by 3 and up by 2
a reflection across the y-axis and a move to the right by 3 and down by 2
a reflection across the y-axis and a move to the right by 3 and down by 2
a refection across the x-axis and a move to the left by 3 and down by 2
a refection across the x-axis and a move to the left by 3 and down by 2
a refletion across the y-axis and a move to the left by 5 and up by 3
being at A" (2 ,-1), B" (-4,0) and C" (-2,5)?(1 point)
Responses
a rotation of - 270 degrees and a move right by 3 and up by 2
a rotation of - 270 degrees and a move right by 3 and up by 2
a reflection across the y-axis and a move to the right by 3 and down by 2
a reflection across the y-axis and a move to the right by 3 and down by 2
a refection across the x-axis and a move to the left by 3 and down by 2
a refection across the x-axis and a move to the left by 3 and down by 2
a refletion across the y-axis and a move to the left by 5 and up by 3
Answers
GPT-4o mini
To find the composition transformation that results in triangle \( \triangle A"B"C" \) being at the points \( A"(2, -1) \), \( B"(-4, 0) \), and \( C"(-2, 5) \), we start by examining the original triangle \( \triangle ABC \) with vertices \( A(1, 1) \), \( B(7, 2) \), and \( C(5, 7) \).
### Step 1: Identify the transformation needed for each point
1. **Point A to A"**:
- Original: \( A(1, 1) \) to \( A"(2, -1) \)
- Change: \( (1 + 1, 1 - 2) = (2, -1) \)
2. **Point B to B"**:
- Original: \( B(7, 2) \) to \( B"(-4, 0) \)
- Change: \( (7 - 11, 2 - 2) = (-4, 0) \)
3. **Point C to C"**:
- Original: \( C(5, 7) \) to \( C"(-2, 5) \)
- Change: \( (5 - 7, 7 - 2) = (-2, 5) \)
### Step 2: Analyze the transformations
From the transformations above:
- For \( A \) to \( A" \): Increase x-coordinate by 1, decrease y-coordinate by 2.
- For \( B \) to \( B" \): Change both x- and y-coordinates, resulting in shifting left by 11 and down by 2.
- For \( C \) to \( C" \): Shift left by 7 in x and down by 2 in y.
It appears that the points are not consistent under a single transformation, meaning it may involve a rotation or reflection followed by translations.
### Step 3: Consider the reflections
Let's explore if any of the answers provided reflects these changes.
- **Reflection across the y-axis**:
- A(1, 1) transforms to A'(-1, 1)
- B(7, 2) transforms to B'(-7, 2)
- C(5, 7) transforms to C'(-5, 7)
This does not align with any target points.
- **Reflection across the x-axis**:
- A(1, 1) transforms to A'(1, -1)
- B(7, 2) transforms to B'(7, -2)
- C(5, 7) transforms to C'(5, -7)
Again, this does not match.
### Step 4: Check the transformations and translations
At this point, we suspect that a reflection might not be necessary. The translation from A, with changes being uniformly spread both in x and y coordinates indicates vectors needing adjustment overall.
### Conclusion
To find the composition transformation to change the triangle as desired, it looks like the transformations pointed out indicate either rotation or reflection combined with moves that are varying significantly for each point.
The best fitting transformation is likely the **reflection across the y-axis followed by translation**, made clear by considering distance adjustments comprehensively.
Among the provided options, without the exact reflection specifics and internal checks through angles and axes transformation:
The answer is likely the **reflection across the y-axis and a move to the left by 5 and up by 3**.
Based on this, option 4 appears to fit:
**a reflection across the y-axis and a move to the left by 5 and up by 3**.
### Step 1: Identify the transformation needed for each point
1. **Point A to A"**:
- Original: \( A(1, 1) \) to \( A"(2, -1) \)
- Change: \( (1 + 1, 1 - 2) = (2, -1) \)
2. **Point B to B"**:
- Original: \( B(7, 2) \) to \( B"(-4, 0) \)
- Change: \( (7 - 11, 2 - 2) = (-4, 0) \)
3. **Point C to C"**:
- Original: \( C(5, 7) \) to \( C"(-2, 5) \)
- Change: \( (5 - 7, 7 - 2) = (-2, 5) \)
### Step 2: Analyze the transformations
From the transformations above:
- For \( A \) to \( A" \): Increase x-coordinate by 1, decrease y-coordinate by 2.
- For \( B \) to \( B" \): Change both x- and y-coordinates, resulting in shifting left by 11 and down by 2.
- For \( C \) to \( C" \): Shift left by 7 in x and down by 2 in y.
It appears that the points are not consistent under a single transformation, meaning it may involve a rotation or reflection followed by translations.
### Step 3: Consider the reflections
Let's explore if any of the answers provided reflects these changes.
- **Reflection across the y-axis**:
- A(1, 1) transforms to A'(-1, 1)
- B(7, 2) transforms to B'(-7, 2)
- C(5, 7) transforms to C'(-5, 7)
This does not align with any target points.
- **Reflection across the x-axis**:
- A(1, 1) transforms to A'(1, -1)
- B(7, 2) transforms to B'(7, -2)
- C(5, 7) transforms to C'(5, -7)
Again, this does not match.
### Step 4: Check the transformations and translations
At this point, we suspect that a reflection might not be necessary. The translation from A, with changes being uniformly spread both in x and y coordinates indicates vectors needing adjustment overall.
### Conclusion
To find the composition transformation to change the triangle as desired, it looks like the transformations pointed out indicate either rotation or reflection combined with moves that are varying significantly for each point.
The best fitting transformation is likely the **reflection across the y-axis followed by translation**, made clear by considering distance adjustments comprehensively.
Among the provided options, without the exact reflection specifics and internal checks through angles and axes transformation:
The answer is likely the **reflection across the y-axis and a move to the left by 5 and up by 3**.
Based on this, option 4 appears to fit:
**a reflection across the y-axis and a move to the left by 5 and up by 3**.