A composition of transformations maps $\triangle XYZ$ to $\Delta X^{"'} Y^{"'} Z^{"''}$. The first transformation for this composition is
A. a $180^{\circ}$ rotation about point X
B. a $270^{\circ}$ rotation about point $X$
C. a translation to the right
D. a reflection across line $m$
and the second transformation is a $90^{\circ}$ rotation about point $X^{\'}$.
Answer (1)
Analyzed each possible first transformation: 18 0 ∘ rotation, 27 0 ∘ rotation, translation, and reflection.
Determined that all options are valid without additional information.
Considered simplicity and chose the 18 0 ∘ rotation about point X as the first transformation.
Concluded that the first transformation is a 18 0 ∘ rotation about point X .
Explanation
Problem Analysis Let's analyze the given transformations and determine which one could be the first transformation in the composition that maps triangle X Y Z to triangle X ′′ Y ′′ Z ′′ . The second transformation is a 9 0 ∘ rotation about point X ′ , where X ′ is the image of point X after the first transformation. We need to consider each option for the first transformation and see if it's a valid possibility.
Case 1: 180 degree rotation Case 1: The first transformation is a 18 0 ∘ rotation about point X . In this case, X ′ = X . The second transformation is a 9 0 ∘ rotation about X . This is a valid composition of transformations.
Case 2: 270 degree rotation Case 2: The first transformation is a 27 0 ∘ rotation about point X . In this case, X ′ = X . The second transformation is a 9 0 ∘ rotation about X . This is also a valid composition of transformations.
Case 3: Translation to the right Case 3: The first transformation is a translation to the right. In this case, X ′ is to the right of X . The second transformation is a 9 0 ∘ rotation about X ′ . This is a valid composition of transformations.
Case 4: Reflection across line m Case 4: The first transformation is a reflection across line m . In this case, X ′ is the reflection of X across line m . The second transformation is a 9 0 ∘ rotation about X ′ . This is a valid composition of transformations.
Final Answer Since all the given options are valid first transformations, we cannot determine a single correct answer without more information. However, the question implies that only one of the options is correct. Let's re-evaluate the options and consider if there's a subtle constraint we missed. The problem states 'The first transformation for this composition is'. This suggests there's a specific composition in mind. Without additional information or context (e.g., a diagram or specific coordinates), we cannot definitively choose one transformation over the others. However, since the problem is multiple choice, and we are forced to pick one, let's consider the 'simplest' transformation. A rotation about a point is generally considered simpler than a translation or reflection. Between a 180 and 270 degree rotation, a 180 degree rotation is arguably simpler. Therefore, let's choose a 180 degree rotation about point X.
Conclusion The first transformation is a 18 0 ∘ rotation about point X .
Examples
Understanding transformations is crucial in computer graphics and animation. For instance, rotating a character by 180 degrees, then rotating it again by 90 degrees can create a specific pose or movement. These transformations are fundamental in creating realistic animations and interactive experiences in video games and movies.
