To determine which pairs of triangles can be mapped to each other using both a translation and a rotation about the common point C, we need to examine each case:
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Triangles X Y C and A B C (both congruent, sharing C): Since both triangles share the point C and are congruent, they can be mapped to each other through a combination of translation (to align them) and rotation around point C. This means this pair can indeed be mapped.
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Triangles X Y Z and A B C (congruent, X Y Z is higher): If triangle X Y Z is identical to triangle A B C but is only positioned higher, we can simply translate triangle A B C down to meet triangle X Y Z and then, if needed, we could rotate to align them further. Therefore, this pair can also be mapped.
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Triangles X Y Z and A B C (congruent, X Y Z reflected to form A B C): Since triangle A B C is formed by reflecting triangle X Y Z, this indicates they can't be aligned purely by translation and rotation. Therefore, this pair cannot be mapped.
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Triangles X Y Z and A B C (congruent, X Y Z rotated down and to the left): In this case, if triangle X Y Z is rotated and is also described as being positioned slightly higher than triangle A B C, it suggests there's also a translation needed to align them correctly, after the rotation. Thus, this pair can be mapped as well.
In summary, the triangle pairs that can be mapped to each other using both a translation and a rotation about point C are:
- Triangles X Y C and A B C
- Triangles X Y Z and A B C (identical but higher)
- Triangles X Y Z and A B C (rotated down and to the left)
The pair that cannot be mapped is:
- Triangles X Y Z and A B C (reflected).