Which of the four figures is the pre-image of triangle ABC before any transformation happened?
Understand the Problem
The question is asking to identify the original position of triangle ABC before it was transformed through reflection and translation. We need to analyze the given options to determine which figure reflects the pre-image of the triangle prior to transformations.
Answer
Option C.
Answer for screen readers
The original position of triangle ABC before any transformations is option C.
Steps to Solve
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Identify the final position of triangle ABC
The triangle shown as $A'B'C'$ has been reflected over both axes and translated. We need to determine the coordinates of the original triangle ABC before these transformations. -
Determine the reflection transformation
Reflecting a point $(x, y)$ over the x-axis results in $(x, -y)$ and over the y-axis results in $(-x, -y)$. Thus, we must reverse these transformations to find the coordinates of triangle ABC. -
Reverse the transformation
The coordinates of $A'B'C'$ can be identified as follows:- If $A'$ is at $(2, 6)$, reflecting back gives:
- Reflect over y-axis: $A( -2, 6)$
- Then reflect over x-axis: $A( -2, -6)$
- For $B'$ at $(4, 2)$:
- Reflect over y-axis: $B( -4, 2)$
- Then react over x-axis: $B( -4, -2)$
- For $C'$ at $(2, 4)$:
- Reflect over y-axis: $C( -2, 4)$
- Then react over x-axis: $C( -2, -4)$
- If $A'$ is at $(2, 6)$, reflecting back gives:
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Identify translation
The original triangle was then translated. We need to determine the translation that would have resulted in the given positions. This typically involves identifying a vector that was applied to each vertex. -
Select pre-image
Using the transformed coordinates, analyze the provided options (A, B, C, D) to find the triangle with the coordinates derived from reversing the transformations.
The original position of triangle ABC before any transformations is option C.
More Information
Triangle transformations involve reflections, translations, and possibly rotations. By reversing these transformations systematically, we can identify the original coordinates needed to reflect them correctly before the transformations occurred.
Tips
- Ignoring order of transformations: Always apply transformations in reverse order to determine original positions.
- Misidentifying coordinates after reflection: Remember that reflection over the axes changes both the x and y coordinates.
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