Question
Describe in words how to map Upper DeltaABC to its image Upper DeltaA'B'C'.
-6
6
0
6
x
y
A
B
C
A'
B'
C'
G
H
J
A horizontal x-axis is labeled from negative 6 to 6. A vertical y-axis is labeled from 0 to 6. There are two triangles plotted. Triangle A B C has the following vertices: A, (negative 4, 4); B, (negative 2, 3); C, (negative 5, 2). Triangle A prime B prime C prime has the following vertices: A prime, (4, 4); B prime, (2, 3); C prime, (5, 2).
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.
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Question content area right
Part 1
Which words below describe the mapping?
A.
Upper DeltaA'B'C' is the image of Upper DeltaABC after a reflection across the line x equals negative 2.
B.
Upper DeltaA'B'C' is the image of Upper DeltaABC translated 2 units horizontally.
C.
Upper DeltaA'B'C' is the image of Upper DeltaABC after a reflection across the x-axis.
D.
Upper DeltaA'B'C' is the image of Upper DeltaABC after a reflection across the y-axis.
-6
6
0
6
x
y
A
B
C
A'
B'
C'
G
H
J
A horizontal x-axis is labeled from negative 6 to 6. A vertical y-axis is labeled from 0 to 6. There are two triangles plotted. Triangle A B C has the following vertices: A, (negative 4, 4); B, (negative 2, 3); C, (negative 5, 2). Triangle A prime B prime C prime has the following vertices: A prime, (4, 4); B prime, (2, 3); C prime, (5, 2).
.
.
.
Question content area right
Part 1
Which words below describe the mapping?
A.
Upper DeltaA'B'C' is the image of Upper DeltaABC after a reflection across the line x equals negative 2.
B.
Upper DeltaA'B'C' is the image of Upper DeltaABC translated 2 units horizontally.
C.
Upper DeltaA'B'C' is the image of Upper DeltaABC after a reflection across the x-axis.
D.
Upper DeltaA'B'C' is the image of Upper DeltaABC after a reflection across the y-axis.
Answers
GPT-4o mini
To map triangle \(\Delta ABC\) to its image \(\Delta A'B'C'\), we need to analyze the transformation applied to each vertex of \(\Delta ABC\).
1. **Original Vertices** of triangle \(\Delta ABC\):
- \(A (-4, 4)\)
- \(B (-2, 3)\)
- \(C (-5, 2)\)
2. **Image Vertices** of triangle \(\Delta A'B'C'\):
- \(A' (4, 4)\)
- \(B' (2, 3)\)
- \(C' (5, 2)\)
### Analyzing the Transformation
Looking at the x-coordinates of the vertices:
- \(A\) at \((-4, 4)\) maps to \(A'\) at \((4, 4)\): The x-coordinate changes from \(-4\) to \(4\), which is a reflection across the y-axis (\(x = 0\)).
- \(B\) at \((-2, 3)\) maps to \(B'\) at \((2, 3)\): Again, the x-coordinate changes from \(-2\) to \(2\), consistent with a reflection across the y-axis.
- \(C\) at \((-5, 2)\) maps to \(C'\) at \((5, 2)\): The same reflection is observed from \(-5\) to \(5\).
### Conclusion
Since all vertices change their x-coordinates from negative to positive while keeping their y-coordinates the same, we conclude that the transformation maps triangle \(\Delta ABC\) to triangle \(\Delta A'B'C'\) as follows:
**The correct description is:**
**D. \(\Delta A'B'C'\) is the image of \(\Delta ABC\) after a reflection across the y-axis.**
1. **Original Vertices** of triangle \(\Delta ABC\):
- \(A (-4, 4)\)
- \(B (-2, 3)\)
- \(C (-5, 2)\)
2. **Image Vertices** of triangle \(\Delta A'B'C'\):
- \(A' (4, 4)\)
- \(B' (2, 3)\)
- \(C' (5, 2)\)
### Analyzing the Transformation
Looking at the x-coordinates of the vertices:
- \(A\) at \((-4, 4)\) maps to \(A'\) at \((4, 4)\): The x-coordinate changes from \(-4\) to \(4\), which is a reflection across the y-axis (\(x = 0\)).
- \(B\) at \((-2, 3)\) maps to \(B'\) at \((2, 3)\): Again, the x-coordinate changes from \(-2\) to \(2\), consistent with a reflection across the y-axis.
- \(C\) at \((-5, 2)\) maps to \(C'\) at \((5, 2)\): The same reflection is observed from \(-5\) to \(5\).
### Conclusion
Since all vertices change their x-coordinates from negative to positive while keeping their y-coordinates the same, we conclude that the transformation maps triangle \(\Delta ABC\) to triangle \(\Delta A'B'C'\) as follows:
**The correct description is:**
**D. \(\Delta A'B'C'\) is the image of \(\Delta ABC\) after a reflection across the y-axis.**