To determine if triangles \( \triangle ABC \) can be mapped onto \( \triangle EFD \) through transformations, we need to analyze the options provided:
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Dilation: This transformation changes the size of the triangle but does not affect the angles. It is often necessary if the two triangles are of different sizes but similar in shape.
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Rotation: This transformation can change the orientation of the triangle without altering its shape or size.
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Translation: This transformation slides the triangle in a specific direction but does not change its shape or size.
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Reflection: This transformation flips the triangle over a line (creating a mirror image) and does not alter the size or angles of the triangle.
To show that \( \triangle ABC \cong \triangle EFD \) using transformations, it often requires a combination of these transformations depending on their initial positions, sizes, and orientations. The key transformations involved could be:
- Dilation (if the triangles are of different sizes)
- Rotation (to match the orientation)
- Translation (to align the position)
- Reflection (if a mirror image is needed to match orientations)
Answer:
- Dilation: If sizes differ.
- Rotation: To align orientation.
- Translation: To adjust position.
- Reflection: If needed to align orientations.
Question 2:
For the theorem that justifies the congruence after the transformations, you would select:
- SAS (Side-Angle-Side), if you have a side from one triangle equal to a side from the other triangle and the included angle between those sides is also equal.
Depending on the specific conditions and descriptions of triangles \( \triangle ABC \) and \( \triangle EFD \), you could justify the congruence using one of the other criteria as well, but given the context in which transformations lead to congruence, SAS is a common and valid choice.
Thus, the response to Question 2 might be SAS, provided that the transformations preserve the side lengths and angles appropriately.