To determine the type of transformation from triangle ABC to A'B'C', we need to analyze the coordinates of the points.
We have the following points:
- \( A = (0, 5) \)
- \( B = (5, 7) \)
- \( C = (7, 0) \)
And their corresponding points after transformation:
- \( A' = (0, 2.5) \)
- \( B' = (2.5, 3.5) \)
- \( C' = (3.5, 0) \)
Dilation
You mentioned that besides dilation, we need to identify other transformations that could have occurred. The transformation of coordinates shows a consistent factor of scaling between the original points and their images. Observing the coordinates, it appears each point in triangle ABC is scaled down by a factor of \(0.5\) (i.e., half the distance from the origin in both x and y directions).
Additional Transformations
However, scaling alone doesn't give us the whole picture.
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Translation: Translating points means moving them by a certain amount in the x and/or y direction. In this case, there does not appear to be a consistent translation since the points have been scaled from a distance rather than moved uniformly.
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Reflection: Reflecting points can create corresponding images across a line. The coordinates do not support any reflection in this case.
Observation of Transformation
It is indeed a dilation, but the relationship between the points suggests a linear transformation (which includes dilation), whereby:
- Each point in triangle ABC is multiplied by \(0.5\):
- \( A' = (0 \times 0.5, 5 \times 0.5) = (0, 2.5) \)
- \( B' = (5 \times 0.5, 7 \times 0.5) = (2.5, 3.5) \)
- \( C' = (7 \times 0.5, 0 \times 0.5) = (3.5, 0) \)
Conclusion
This is primarily a dilation, with a scale factor of \(0.5\) centered at the origin. Other than purely dilation, it can be categorized under linear transformations, as they encompass scaling and preserving the lines of the shapes, thereby indicating no additional transformations such as rotation or reflection are combined here.
In summary, besides dilation, the transformation can be described as a linear transformation that preserves the shape of triangle ABC while altering its size.