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Given and , and that is the preimage of , represent the transformation algebraically.
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Q. Given and , and that is the preimage of , represent the transformation algebraically.
- Compare Coordinates: To find the transformation, we need to compare the coordinates of the corresponding points in triangles and . Let's start by examining the changes in the -coordinates.
- Analyze X-Coordinates: The x-coordinate of point C is , and the x-coordinate of point U is . The change in x is . The x-coordinate of point D is , and the x-coordinate of point V is . The change in x is . The x-coordinate of point E is , and the x-coordinate of point W is . The change in x is .
- Check Y-Coordinates: The changes in the x-coordinates are not consistent , which suggests that there might be a mistake or that the transformation involves more than just a translation. We need to check the y-coordinates to understand the transformation better.
- Calculate Distances: The -coordinate of point is , and the -coordinate of point is . The change in is . The -coordinate of point is , and the -coordinate of point is . The change in is . The -coordinate of point is , and the -coordinate of point is . The change in is .
- Identify Dilation Factor: The changes in the -coordinates are also not consistent , which confirms that the transformation is not a simple translation. We need to consider other types of transformations such as rotation, reflection, dilation, or a combination of these.
- Find Center of Dilation: Let's analyze the distances between the points in each triangle to determine if there is a dilation (scaling) factor. We will calculate the distance between points and and compare it to the distance between points and .
- Adjust Transformation: The distance between points and is . The distance between points and is .
- Consider Translation: The ratio of the distances is . This suggests that there is a dilation by a factor of centered at some point. However, we need to determine the center of dilation and whether there are additional transformations involved.
- Identify Error: To find the center of dilation, we can look for a point that maintains its position relative to the other points after the transformation. Since the -coordinate of point and point are both , and the -coordinates have been halved, it suggests that point could be the center of dilation.
- Identify Error: To find the center of dilation, we can look for a point that maintains its position relative to the other points after the transformation. Since the -coordinate of point E and point W are both , and the -coordinates have been halved, it suggests that point E could be the center of dilation.If point E is the center of dilation, then the transformation of any point would be to to account for the scaling factor of . However, this does not account for the translation observed in the -coordinates. We need to adjust our transformation to include this translation.
- Identify Error: To find the center of dilation, we can look for a point that maintains its position relative to the other points after the transformation. Since the -coordinate of point E and point W are both , and the -coordinates have been halved, it suggests that point E could be the center of dilation.If point E is the center of dilation, then the transformation of any point would be to to account for the scaling factor of . However, this does not account for the translation observed in the -coordinates. We need to adjust our transformation to include this translation.Considering the translation in the -coordinates, we can see that point C has moved from to , which is a translation of in the -direction and a scaling of in the -direction. Similarly, point D has moved from to , which is a translation of in the -direction and a scaling of in the -direction.
- Identify Error: To find the center of dilation, we can look for a point that maintains its position relative to the other points after the transformation. Since the -coordinate of point E and point W are both , and the -coordinates have been halved, it suggests that point E could be the center of dilation.If point E is the center of dilation, then the transformation of any point would be to to account for the scaling factor of . However, this does not account for the translation observed in the -coordinates. We need to adjust our transformation to include this translation.Considering the translation in the -coordinates, we can see that point C has moved from to , which is a translation of in the -direction and a scaling of in the -direction. Similarly, point D has moved from to , which is a translation of in the -direction and a scaling of in the -direction.The inconsistency in the translation of the -coordinates suggests that there might be an error in the problem statement or that the transformation is not a simple dilation and translation. Without consistent transformations for all points, we cannot represent the transformation algebraically as requested.
